WEBVTT mathematics/pre-calculus/selhorst-jones
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Hi--welcome back to Educator.com.
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Today, we are going to talk about limits at infinity and limits of sequences.
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Up until this point, we have only considered the idea of a limit as x goes to c, where c is some fixed horizontal location.
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But what if, instead of focusing on a single location, we considered what would happen to the function if x just kept traveling forever off to infinity?
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What would we do if our x wasn't going to a single place, but we were just sort of watching it ride off into the sunset?
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This is the idea that we will consider in this lesson.
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In fact, we have actually considered this many lessons ago, in the lesson on horizontal asymptotes,
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because a horizontal asymptote was the question of what this function goes to in the very, very long run;
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as x runs off to infinity (both the positive and the negative infinity)
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what vertical value, what y-value, what function value, do we end up running to in the long run?
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That was the idea of a horizontal asymptote; we will draw on those ideas in this lesson, as well.
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Before we start, though, let's remind ourselves of something important: infinity is not a location.
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It is not some place; you don't run to the end of the rainbow, and there you are at infinity; you always have to keep running.
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The idea of infinity is just going on forever, moving on to ever-larger numbers.
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You never actually reach infinity; you can travel towards infinity, but you can never reach infinity.
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It is just the idea of going on forever; it is the idea of riding into the sunset.
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You don't ever actually make it to the sun; you just keep riding off into the distance.
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That is the idea of what it means to travel towards infinity.
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You never actually arrive there; x can't be equal to infinity; but we can consider the idea of what happens as x runs off forever and ever and ever.
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That is what we will be thinking about.
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We denote the limit of a function at infinity (even though it says "at," we are really meaning "as it goes towards")--
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the limit as x goes to negative infinity of f(x) and the limit as x goes to positive infinity of f(x)...
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And this means the value that f(x) approaches as x goes off to either negative infinity or positive infinity, respectively.
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So, negative infinity, with an actual negative sign--that means negative infinity.
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And infinity, just on its own with nothing there--we just assume that there is a positive sign in front of it, even though we don't see it.
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If you don't see a symbol, it is assumed that we are talking about positive infinity, as opposed to talking about negative infinity.
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Negative infinity would go off to the left, whereas positive infinity, or just infinity with no symbol in it, goes off to the right, forever.
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OK, a limit at infinity works very similarly to how a normal limit works.
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Does the function settle down--does it go to some specific value l?
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It is just that, in this case, we are talking about long-term behavior, instead of x going to some specific horizontal location.
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Instead of what happens to the function as x gets close to c, it is what happens to the function as it rides off into the sunset.
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What happens as it goes off to some infinity?
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Now, it is important to note that most of the functions--in fact, the vast majority of the functions
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that we are used to working with--do not have limits at infinity.
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For example, if we consider f(x) = x, one of the simplest functions that we are used to using,
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that one just keeps growing forever; so it has no limit at infinity.
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It doesn't stabilize; it doesn't settle down to some value.
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If you plug in 1 million, you get 1 million out of it; if you plug in 1 billion, you get 1 billion out of it; if you plug in 1 trillion, you get 1 trillion out of it.
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As you keep plugging in larger and larger numbers, it is just going to keep growing and growing and growing and growing.
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It is never going to stop growing, which means it is never going to settle down, which means it is never going to some specific value l.
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Most of the functions that we are used to dealing with on a daily basis are actually not going to have limits at infinity,
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because they never settle down to a specific value--they grow without bound.
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They might grow off to positive infinity; they might grow off to negative infinity (that is, vertically--what the output goes off to be).
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But they are growing without bound; they aren't going to some specific value, so that means that they won't have limits at infinity.
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Still, there are definitely functions that do have limits at infinity.
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The type of functions that we will work with most often (there are some others that won't be this, but)
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the ones that we will work with most often, that have limits at infinity, are rational functions.
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We worked with these many lessons ago, when we learned about asymptotes.
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They are functions of the form f(x) = n(x)/d(x), where n(x) and d(x) are polynomials, and d(x) is not equal to 0.
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So, you probably remember these things like this: g(x) = (3x - 1)/(x³ + 4), h(x) = 1/x⁴,
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j(x) = (x⁵ + 47x²)/(x³ - 15)--just some polynomial divided by some other polynomial.
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Now, because we are dividing by something, that means that our denominator, what we are dividing by,
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has the possibility to grow faster than the numerator.
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Basically, our denominator can grow fast enough to keep the numerator in check--to keep that numerator from blowing off and just going on forever.
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The denominator can actually grow faster and keep it down.
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It has the ability to stabilize it to a single value in the long run.
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And that is why we end up seeing rational functions give us limits at infinity so often.
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For a rational function, the question is basically comparing the long-term growth rates of the numerator and the denominator.
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It is a question of which is growing faster over the long term: is it the numerator or the denominator?
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If the numerator is growing faster than the denominator over the long term, then the thing is not really going to settle down,
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because it is just going to keep getting bigger and bigger and bigger.
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If, on the other hand, the denominator is growing faster than the numerator,
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then the denominator will crush the numerator, and so it will be forced to settle down.
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Now, we already studied this idea in horizontal asymptotes; and so, let's look at those results.
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If we have some rational function, f(x) (and notice that that is just some polynomial divided by some polynomial--
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some constant times x^n, some other constant times x^n - 1, and working our way
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down to a constant times x, plus some constant, and the same thing on the bottom, as well--
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it is just constant times x to some value, constant times x to that value minus 1,
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and working our way down to a constant; so it is some polynomial over some polynomial),
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notice, from this, that n is the numerator's degree; we have x^n as the largest exponent on the top;
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and m is the denominator's degree, so m is the biggest exponent on the bottom.
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From this, there are three possibilities: if n is less than m, then that means that our top,
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the exponent in our numerator, n, is going to be less than the exponent in our denominator,
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which means that the denominator is going to grow faster, so it will crush the numerator, causing us to have a horizontal asymptote of y = 0.
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If, on the other hand, n equals m (the leading exponent on the numerator is equal
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to the leading exponent on the denominator), they grow in the same category of speed.
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They won't necessarily have precisely the same; but one of them won't massively outclass the other one.
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At that point, what we will do is compare the leading coefficients, a<font size="-6">n</font> and b<font size="-6">m</font>.
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The horizontal asymptote that we get out of that is a ratio of the leading coefficients, a<font size="-6">n</font> divided by b<font size="-6">m</font>,
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because in the long run, since we have the same exponent on top and bottom,
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that part, the x to the some exponent, will grow at the same rate on the top and the bottom.
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So, it will end up just being a question of what number they are multiplying in front.
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And that is why we get a horizontal asymptote based on that.
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And finally, the last one: if n is greater than n (that is, the leading exponent on the numerator is greater
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than the leading exponent on our denominator), then that means that the numerator will be able to run faster
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than the denominator and escape the denominator's ability to bound it and hold it back.
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And so, it will just go off forever, and it won't be able to stabilize to a single value, which means that it will have no horizontal asymptote.
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We can write this in a way where we can talk about this as limits at infinity, because,
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since horizontal asymptotes tell us the behavior of f(x) as x goes to positive or negative infinity--
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a horizontal asymptote is what value it approaches over the long term--they are also telling us the limits at infinity,
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since the limit at infinity is what value it approaches over the long term.
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So, for some rational function f(x), let n be the numerator's degree, and m be the denominator's degree.
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Let a<font size="-6">n</font> and b<font size="-6">m</font> be the leading coefficients of the numerator and the denominator, respectively.
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Then, we have: if n is less than m, the limit as x goes to positive or negative infinity of f(x) equals 0.
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The numerator is smaller, effectively; the exponent is smaller, so its growth rate is smaller than the denominator.
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So, the denominator crushes it down to 0.
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If n equals m, then the limit as x goes to positive or negative infinity of f(x) is equal to a<font size="-6">n</font> divided by b<font size="-6">m</font>.
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The growth rate on the top and the bottom is the same, because they have the same leading exponent.
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So now, it is a question of what the ratio is of the coefficients in front of them.
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And finally, if n is greater than m, that means that the leading coefficient on top is greater than the leading coefficient on the bottom,
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which means that the growth rate of the top is greater than the growth rate of the bottom.
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So, there the top manages to escape and run off forever.
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So, that means that the limit as x goes to positive or negative infinity of f(x) simply does not exist, because it will never stabilize to a single value.
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That tells us what to do with rational functions; the previous method allows us to easily find limits at infinity for rational functions.
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But we will occasionally have to deal with other types of functions, as well.
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We won't only have to deal with rational functions.
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So, in that case, the best thing to do--there is no simple formulaic method for how to figure out "Here is what the limit is going to end up being."
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In this case, what you want to do is think in terms of how the function will be affected as x grows very large, both positively and negatively.
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You want to think, "Does the function grow without bound?"
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If it just grows forever and ever, or goes off down forever and ever,
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then that means that it is going to end up not stabilizing to something in the long term, which means that it won't have a limit at infinity.
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Or, on the other hand, will it settle down--does it go to some specific value?
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Does it settle down; does it approach some specific thing over the long term?
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So, there are two good ways to think about this, to figure out if this is the case.
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We want to think about what happens if we plug in a large number.
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And by "large number," I mean to think like...if I were to plug in something on the scale of a million,
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a billion, a trillion, a really, really, really big number--something large--what would happen to this?
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You don't have to come up with actual answers to what will happen to the thing.
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You don't have to produce some number in the end.
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You just want to think, "If there was a really, really big number here, how would it affect the other things that it is near?"
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What would no longer really be important? What would still matter?
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If you were to plug in a really big number, what is going to keep changing?
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And what will happen as that large number continues to increase?
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That is one way of looking at it; the other way to look at it is to think, "What are the rates of growth in the function?"
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Which part of the function is growing faster and will continue to grow?
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Which parts grow faster, and which parts are growing slower, that get slower and slower as we go farther and farther out?
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Thinking about these two things (that one will especially help if you are dealing with a fraction--
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what is the comparison between the growth rate of our numerator versus the growth rate of our denominator?)--
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thinking in terms of rates of growth, what is growing faster and what is growing slower,
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how the rate of growth will be affected as we go out to larger and larger x--
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these sorts of things (that and what would happen if I plugged in a very large number)--
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thinking in terms of those two ideas will give you a good, intuitive sense of what is going to happen.
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If you think through these questions, you can get a good idea of where the function will be going,
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of how the function will behave over the long term.
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You won't necessarily be able to come up with an absolutely precise answer.
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But you will be able to get a sense of if it makes sense for this thing to have a limit at all.
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But sometimes, you will even be able to get an exact answer by thinking through this; it depends on the situation.
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But just sort of try to be creative and think in a very broad, general sense.
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We will also talk about ways where, if you are not quite sure how to think about it,
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there are numerical ways that you can figure out...get a good sense of what is going on.
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And we will talk about that in just a few slides.
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Another thing that we can talk about, using this idea of a limit at infinity, as a limit goes to infinity: we can apply it to a sequence, as well.
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If we have some sequence, a₁, a₂, a₃, a₄, a₅...
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so it is some infinite sequence that just keeps going on forever and ever and ever--
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then what we can consider is the limit as n goes to infinity of a<font size="-6">n</font>.
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What does our sequence go towards--what value does the sequence approach in the long run?
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How does this thing work out? What will it be going towards as n becomes ever larger and larger and larger?
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The limit of a sequence (that is this thing right here) is very similar to the limit of a function at infinity.
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The question is, "Does the sequence settle down--does it go to some specific value l as n runs off to infinity?"
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As our n becomes larger and larger and larger, does our sequence stabilize
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into something that is going to basically be the same as we go farther and farther and farther?
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Now, it is important to note that, just like functions, most sequences will not have limits as n goes to infinity.
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For example, a really simple sequence: 1, 2, 3, 4, 5, 6... has no limit, because it just grows forever.
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It will just grow forever, because we have 1, 2, 3, 4...so if we look at a very far-out term, it will be very large.
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But if we look at an even farther-out term, it will have continued to grow, and it will be even larger.
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So, it is not going to head towards a steady value; it is not going to stabilize and go to some specific value l; it will never settle down.
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Nonetheless, there are definitely still some sequences out there that will end up stabilizing; and we will see those in the examples.
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But just because we are looking at the limit as n goes to infinity of a sequence doesn't necessarily mean that it will stabilize.
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There are plenty of sequences out there that won't stabilize at all.
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For example, every arithmetic sequence we have ever looked at won't stabilize,
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because it just continues stepping up and stepping up and stepping up and stepping up.
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Finally, we can also talk about numerical evaluation.
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Sometimes, it can be difficult to tell how a function or sequence will behave in the long run.
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In that case, we can evaluate the function numerically--that is to say, use numbers.
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We will just plug in numbers, and we will see what comes out.
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If we have a calculator, we can use a calculator and just plug in very large numbers.
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And we will want to plug in both positive and negative numbers.
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And then, see what happens: we just see what happens to our function or a sequence.
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We plug in 10; then we plug in 100; then we plug in 1000; then we plug in 10000.
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Does it seem like it is going to a number, or does it seem like it is just growing larger and larger and larger?
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Doing this will give us a good sense for long-term behavior.
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We will be able to tell that, yes, it seems to be just growing forever and ever,
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or it seems to be stabilizing as we go to larger and larger numbers that we are plugging in.
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So, this gives us a way to numerically get a sense of what is going on.
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It is not foolproof, but for the most part you will be able to figure out which one it is going to end up going to.
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And you probably also have a very good estimate of what value it is going to be approaching in the long term.
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Similarly, if you have access to a graphing calculator or some graphing program, you can graph the function.
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If you expand the viewing window to a large horizontal region (say -100 to +100),
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then you can look and see if the graph settles down in the long run.
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Does it seem like it is being pulled to a single value, or does it seem like it is just blowing forever and going to keep growing forever and ever and ever?
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Now, once again, it is not a foolproof method; there are some times where the function will fool you for the first 1000x.
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From 0 to 1000, it will look like it is growing forever and ever.
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But then, after 1000, it will actually end up steadying out to a single value.
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But for the most part, this is a pretty good way to see if this is going to end up approaching a single value,
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or if it is going to just keep growing forever and ever.
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So, just take a look at the graph; make sure you use a large horizontal region.
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If you only look at -10 to +10 for x, you might not have a very good sense of what happens in the long run.
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You want to use a very large horizontal region, like -100 to +100.
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It might be kind of hard for your graphing calculator to graph as you get to larger and larger windows;
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but the larger you can deal with, the better, really, because that will tell you a better idea of what is going on.
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For the most part, though, -100 to 100 should probably do for anything that you want to graph.
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And then, just look: does the graph settle down--it is going to tend to a single value in the long run?
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As you go to those larger x-values, is it basically always graphing at the same height?
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And if that is the case, it probably has a limit; and you can figure out, looking at the graph, about what value that ends up being.
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All right, we are ready for some examples.
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The first example: Evaluate the limits below if they exist.
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The first one is the limit, as x goes to negative infinity, of 1/x.
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In this case, we want to think about what ends up happening.
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We have that specific formulaic method; we have a step-by-step thing for analyzing what is going to come out here.
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We already can see that the answer has to be 0, from that formulaic method.
00:16:58.200 --> 00:17:04.900
But let's also think about what happens: as x goes off to negative infinity, our 1/x...
00:17:04.900 --> 00:17:08.700
well, x continues to grow; how does the numerator grow?
00:17:08.700 --> 00:17:13.400
It doesn't grow; it is just 1--it stays constant; it just sits there as 1 forever and ever and ever.
00:17:13.400 --> 00:17:16.600
But our x continues to get larger and larger and larger.
00:17:16.600 --> 00:17:23.200
We plug in 1000 (well, we plug in -1000, because we are going to negative infinity), and we have 1/-1000;
00:17:23.200 --> 00:17:25.600
we plug in negative 10 thousand, and we have 1 over negative 10 thousand;
00:17:25.600 --> 00:17:30.600
we plug in negative 1 billion, and we have 1 over negative 1 billion.
00:17:30.600 --> 00:17:36.000
We are getting smaller and smaller and smaller; it is crushing down to 0, so we see that it ends up being 0.
00:17:36.000 --> 00:17:42.300
We can also tell that that is going to end up being the case, just because our numerator has a leading exponent of 0.
00:17:42.300 --> 00:17:48.600
It is x⁰, which would only give us 1; and our bottom has x¹, so the bottom has a higher leading exponent;
00:17:48.600 --> 00:17:51.900
so that means that it is going to be crushed down to 0 in the long run.
00:17:51.900 --> 00:17:58.800
Similarly, the limit as x goes to positive infinity of 1/x...well, our bottom is x; it does grow;
00:17:58.800 --> 00:18:01.600
but our top is 1--it just stays the same forever and ever.
00:18:01.600 --> 00:18:07.300
So, that means, as x gets larger and larger and larger, that it is going to crush our entire fraction down to 0.
00:18:07.300 --> 00:18:10.800
So, we end up getting 0 for the limit here.
00:18:10.800 --> 00:18:17.200
Over here, we have the limit as x goes to negative infinity of (x³ + 3)/(x² + x).
00:18:17.200 --> 00:18:22.000
So, notice: in this case, we have 3 as the leading exponent up top.
00:18:22.000 --> 00:18:26.000
That means that we have this growth rate somewhere in the neighborhood of x³.
00:18:26.000 --> 00:18:32.600
But on the bottom, we have a leading exponent of squared; so we have a growth rate somewhere in the neighborhood of x².
00:18:32.600 --> 00:18:39.100
What that means is that the top, in the long run, will end up growing much, much, much faster than our bottom will.
00:18:39.100 --> 00:18:41.900
It is going to end up outrunning the bottom, effectively.
00:18:41.900 --> 00:18:44.900
We can also imagine this: if we were to plug in a very large number,
00:18:44.900 --> 00:18:54.700
then we would have big cubed, plus 3, over big squared, plus big.
00:18:54.700 --> 00:19:01.300
Well, notice: the number here that is the most important is big cubed.
00:19:01.300 --> 00:19:06.700
Big squared is a very large number, but big cubed is going to be even larger.
00:19:06.700 --> 00:19:12.500
There is a huge difference between 10² and 10³; 10 squared is 100, but 10 cubed is 1000.
00:19:12.500 --> 00:19:17.200
So, as we get out to very large numbers, big cubed is going to be massively larger than big squared.
00:19:17.200 --> 00:19:22.600
And similarly, big is just to the 1; so it is practically not going to be anything compared to big squared.
00:19:22.600 --> 00:19:27.000
So, in the long run, the plus 3 doesn't really matter; the big to the 1 doesn't really matter.
00:19:27.000 --> 00:19:31.700
The big squared doesn't really matter, because the biggest thing of all, by far, is big cubed.
00:19:31.700 --> 00:19:35.100
So, that means that we are going to get really, really large numbers up top,
00:19:35.100 --> 00:19:37.900
and nothing else is really going to be of a comparative size.
00:19:37.900 --> 00:19:41.200
So, that means, in the long run, that it is going to blow out to infinity.
00:19:41.200 --> 00:19:46.200
In this case, it will blow out to negative infinity; so that means that it is not stabilizing to a single value.
00:19:46.200 --> 00:19:50.300
So, we say that the limit does not exist, because it is never going to stabilize,
00:19:50.300 --> 00:19:54.700
because our top grows faster than our bottom will.
00:19:54.700 --> 00:19:58.700
In this case, it is going to negative infinity, so we might think of it as growing down.
00:19:58.700 --> 00:20:02.300
But in either case, it is going larger than the bottom will.
00:20:02.300 --> 00:20:10.100
The final one is the limit as x goes to positive infinity of (8x⁴ + 3x²)/(2x⁴ - 17).
00:20:10.100 --> 00:20:15.600
So, in this case, we see that the important thing is a leading coefficient of 4 on top, and a leading coefficient of 4 on the bottom.
00:20:15.600 --> 00:20:21.900
3x² and -17...as we get to very large numbers, as we go farther and farther out towards infinity,
00:20:21.900 --> 00:20:27.600
3x² and -17...they aren't really going to matter much in the long run, as we get to very large numbers.
00:20:27.600 --> 00:20:30.800
So, it really is determined by 8x⁴/2x⁴.
00:20:30.800 --> 00:20:35.000
In that case, we see that the x⁴ and the x⁴ are going to effectively cancel each other out.
00:20:35.000 --> 00:20:42.000
So, all that we really have left, in the long run, is 8/2; 8/2 simplifies to 4, and there is our answer.
00:20:42.000 --> 00:20:48.500
We can also see this as the leading coefficients, 8 and 2; since we have the leading exponents,
00:20:48.500 --> 00:20:55.800
we just do the leading coefficient on the top, divided by the leading coefficient on the bottom, 8/2; and that is equal to 4; great.
00:20:55.800 --> 00:21:02.400
The next example: Let's look at the limits here--these are limits of sequences, since it is n going to infinity.
00:21:02.400 --> 00:21:07.300
Evaluate the limits below, if they exist: limit as n goes to infinity of 1/n².
00:21:07.300 --> 00:21:11.800
Once again, we see that this is n²; up at the top, it is just 1; it is constant.
00:21:11.800 --> 00:21:16.300
So, our bottom continues to grow and grow and grow and grow, but our top stays the same.
00:21:16.300 --> 00:21:22.500
So, as we divide by larger and larger and larger numbers, it crushes it down to 0, just like the reasoning that we used previously.
00:21:22.500 --> 00:21:30.800
The limit as n goes to infinity of (5n - 1)/(n + 4): well, in this case, we have 5n and n over here.
00:21:30.800 --> 00:21:35.700
The -1 and the 4 don't change as n goes larger and larger and larger.
00:21:35.700 --> 00:21:42.600
In the long run, we have big numbers for n; -1 and 4 are going to basically have no effect on what is going on.
00:21:42.600 --> 00:21:47.400
So, we can think of them as not really mattering, which leaves us with 5n/n in the long run;
00:21:47.400 --> 00:21:51.600
so we are just comparing--what are the two leading coefficients?
00:21:51.600 --> 00:22:02.900
5/1 = 5; and there is the limit as n goes out to infinity, the limit of the sequence--what happens to the sequence in the long run.
00:22:02.900 --> 00:22:06.100
Compare the limits below: which limit exists? Why?
00:22:06.100 --> 00:22:13.700
All right, our first one is the limit as x goes to infinity of sin(x), and our second one is the limit as x goes to infinity of sin(x)/x.
00:22:13.700 --> 00:22:18.600
OK, let's get a sense for what happens to the limit as x goes to infinity of sin(x).
00:22:18.600 --> 00:22:22.600
Well, first let's take a quick graph of how sin(x) behaves.
00:22:22.600 --> 00:22:33.900
We start here at x = 0; it goes up and down and up and down and up and down and up...
00:22:33.900 --> 00:22:37.900
and it just continues in this method forever and ever and ever and ever.
00:22:37.900 --> 00:22:45.900
It never changes this thing of going up/down, up/down, up/down; that is how sin(x) works--it repeats itself over and over forever.
00:22:45.900 --> 00:22:57.300
What that means is that we have it bouncing; we are bouncing between +1 at its maximum and -1, forever.
00:22:57.300 --> 00:23:04.200
We are always going up/down, up/down, up/down, up/down; we never stop bouncing up and down.
00:23:04.200 --> 00:23:11.400
So, if that is the case, since we never stop bouncing up and down, it never settles down to a specific value.
00:23:11.400 --> 00:23:18.800
All right, it is going to always be near the values of +1 and -1 and 0; but it never steadies out to a single thing.
00:23:18.800 --> 00:23:24.700
If we say that it is going to be at 0 in the long run, well, it is going to end up getting away from 0 over and over and over.
00:23:24.700 --> 00:23:37.600
So, it is never settling down; if it never settles down, that means that the limit does not exist; the limit here does not exist.
00:23:37.600 --> 00:23:43.000
What about our other limit, though--the limit as x goes to infinity of sin(x)/x?
00:23:43.000 --> 00:23:56.900
Well, what happens? Once again, sin(x), our top, bounces between -1 and +1 forever.
00:23:56.900 --> 00:24:09.800
OK, but the bottom grows forever; this x right here is going to get larger and larger and larger as x goes off to infinity.
00:24:09.800 --> 00:24:15.500
So, as x goes off to infinity, the bottom will grow forever.
00:24:15.500 --> 00:24:25.000
The top oscillates between +1 and -1, +1 and -1, +1 and -1; but our bottom gets larger and larger and larger: 1, 10, 100, 1000...
00:24:25.000 --> 00:24:29.700
So, since the top never really manages to get very far--it isn't growing without bound--
00:24:29.700 --> 00:24:35.300
it is just bouncing between these two numbers--even at its largest possible values of +1 and -1,
00:24:35.300 --> 00:24:42.900
if we divide that by x out at a billion, x out at a quadrillion...it is going to be crushing it down to these very small numbers.
00:24:42.900 --> 00:24:47.100
Thus, we have that the fraction will end up being crushed.
00:24:47.100 --> 00:24:56.800
The bottom, in the long run, is going to crush the top.
00:24:56.800 --> 00:24:59.800
In the long run, it ends up looking like 0.
00:24:59.800 --> 00:25:15.200
If you want to see what that ends up looking like, what we have is this divide by x...well, 1/x has a graph like this, as it approaches it.
00:25:15.200 --> 00:25:20.400
So, our sin(x) is going to be bouncing between these two possible extremes.
00:25:20.400 --> 00:25:26.500
So, it ends up getting squeezed down, closer and closer and closer and closer to this 0 value.
00:25:26.500 --> 00:25:31.800
And that is why it ends up happening--that is why you end up having this long-term value of 0.
00:25:31.800 --> 00:25:36.800
In sin(x), it bounces up and down forever; it just keeps going up and down and up and down.
00:25:36.800 --> 00:25:44.100
But over here in sin(x) divided by x, this "divide by x," over the long run, pinches it down--keeps it crushed down.
00:25:44.100 --> 00:25:48.600
It starts with these large oscillations; but as it goes farther and farther out, it has to get smaller and smaller,
00:25:48.600 --> 00:25:53.100
because the x, the "divide by x," crushes it down; and so it gets crushed down to a single value.
00:25:53.100 --> 00:26:00.300
It will continue to oscillate, but it is getting closer and closer and closer; it has to stay in this window near this height value of 0.
00:26:00.300 --> 00:26:05.800
And so, since it gets crushed down slowly over time to 0, it effectively just approaches 0 in the long run.
00:26:05.800 --> 00:26:09.600
So, we have a limit as x goes to infinity of 0.
00:26:09.600 --> 00:26:13.900
The fourth example: Compare the limits below. Which exists? Why?
00:26:13.900 --> 00:26:17.300
First, we could just graph this to get a sense of what is going on.
00:26:17.300 --> 00:26:23.800
If we graph this, the limit as x goes to negative infinity of 2^x, and the limit as x goes to positive infinity of 2^x...
00:26:23.800 --> 00:26:26.400
well, if we graph, what does 2^x look like?
00:26:26.400 --> 00:26:30.500
Well, at 0, it is going to be at 1; and then as we go out, it is going to get very large very quickly.
00:26:30.500 --> 00:26:34.200
As we go to the left, it is going to get smaller and smaller and smaller and smaller.
00:26:34.200 --> 00:26:39.900
All right, that is what happens: it will never get past the x-axis, but it is going to end up getting smaller and smaller and smaller.
00:26:39.900 --> 00:26:48.900
If we look at some values, we see that at x = -1, x = -2, x = -3...for this, we would have 2^-1,
00:26:48.900 --> 00:26:55.800
and then 2^-2, and then 2^-3, which would come out to be 1/2, 1/2², so 1/4,
00:26:55.800 --> 00:27:03.800
1/2³, which would be 1/8; so 1/2, 1/4, 1/8...it gets smaller and smaller.
00:27:03.800 --> 00:27:08.500
It is always going to smaller values as x goes off to negative infinity.
00:27:08.500 --> 00:27:13.300
Since it is always going off to smaller and smaller values--that is, values closer and closer to 0--
00:27:13.300 --> 00:27:18.200
as 2 to the negative number becomes very large--it is going to be
00:27:18.200 --> 00:27:22.200
1 over 2 to the very large number, which is going to make a very tiny fraction.
00:27:22.200 --> 00:27:26.400
So, over the long run, it ends up getting crushed down to 0.
00:27:26.400 --> 00:27:30.600
However, if we look at the limit as x goes to positive infinity of 2^x,
00:27:30.600 --> 00:27:35.100
if we look at just the first couple of numbers, 2¹, and then 2²,
00:27:35.100 --> 00:27:41.200
and then 2³ (that is, x = 1, x = 2, x = 3), we would end up getting 2, and then 4, and then 8;
00:27:41.200 --> 00:27:46.700
so it is getting bigger and bigger as it ends up going larger and larger.
00:27:46.700 --> 00:27:51.500
As it gets closer and closer to positive infinity, it will get larger and larger and larger.
00:27:51.500 --> 00:27:56.700
We end up seeing that, since it is going to get larger and larger and larger, it is never going to stabilize to a single value.
00:27:56.700 --> 00:28:01.700
It is never going to go to some specific value l, so that means that the limit does not exist,
00:28:01.700 --> 00:28:06.400
because it will just blow off forever and ever, going up forever and ever.
00:28:06.400 --> 00:28:16.400
The fifth example: Evaluate the limit as x goes to infinity of 2x/(x + 1) - x²/4(x + 1)².
00:28:16.400 --> 00:28:25.200
The first thing to notice here is that this portion of this fraction here doesn't really have an effect on this fraction here during the process of the limit.
00:28:25.200 --> 00:28:31.100
So, as x goes to infinity, this fraction and this fraction don't really interact with each other.
00:28:31.100 --> 00:28:36.400
They are basically separate; so if they are basically separate, we can split the limit into the two portions.
00:28:36.400 --> 00:28:44.800
So, we can split it into the limit as x goes to infinity of the first portion, 2x/(x + 1),
00:28:44.800 --> 00:28:56.000
minus the limit as x goes to infinity of the second portion, x²/4(x + 1)².
00:28:56.000 --> 00:28:58.100
All right, now we can evaluate both of these on their own.
00:28:58.100 --> 00:29:03.400
For the first one, 2x/x...they both have the same leading coefficient.
00:29:03.400 --> 00:29:08.200
If we imagine very large numbers going in there, we are comparing two times big number, over big number plus 1.
00:29:08.200 --> 00:29:12.600
The plus one doesn't really matter; so we only care about two times big over big.
00:29:12.600 --> 00:29:20.200
The "big"s cancel each other out effectively, and we can think of this as just going...it will go to precisely 2 in the long run.
00:29:20.200 --> 00:29:25.000
As x goes off forever and ever, it is going to get closer and closer to 2.
00:29:25.000 --> 00:29:28.500
Minus the limit as x goes to infinity...for this one, we are not quite sure, because...
00:29:28.500 --> 00:29:32.200
let's expand the (x + 1)², although we can see x²
00:29:32.200 --> 00:29:34.800
divided by something that is also going to contain an x².
00:29:34.800 --> 00:29:45.700
So, we should probably be able to see that in the long run, as x goes to positive infinity, we will end up seeing it go to 1/4.
00:29:45.700 --> 00:29:49.500
But let's expand it, so we can see it clearly: the limit as x goes to infinity...
00:29:49.500 --> 00:29:59.700
x² doesn't change on top...divided by 4, times (x + 1)²...that is just equal to x² + 2x + 1.
00:29:59.700 --> 00:30:09.200
So, if we multiply 4 times that expansion of (x + 1)², we get 4x² + 8x + 4.
00:30:09.200 --> 00:30:14.600
So, we still have 2 in front, minus the limit, as x goes to infinity, of this.
00:30:14.600 --> 00:30:17.300
Well, actually, at this point, we don't even need to do another limit,
00:30:17.300 --> 00:30:24.000
because we can see that the top has a leading exponent of x²; the bottom has a leading exponent of squared, as well.
00:30:24.000 --> 00:30:28.400
So, we just compare the coefficients in front, 1 and 4.
00:30:28.400 --> 00:30:34.500
Since we have big number squared up top, divided by 4 times big number squared, plus 8 times big number...
00:30:34.500 --> 00:30:39.600
that is not really going to be much, compared to big squared...plus just plain 4 (that is not going to be much compared to big),
00:30:39.600 --> 00:30:47.200
it is really 1 big squared, over 4 big squared; the big squareds effectively cancel out, leaving us with 1/4 in the long run.
00:30:47.200 --> 00:30:52.100
We have minus 1/4; we have broken down each piece of the limit; we have figured out
00:30:52.100 --> 00:31:02.100
that the first portion becomes 2; the second portion becomes 1/4; so 2 - 1/4 simplifies to 7/4.
00:31:02.100 --> 00:31:11.100
All right, the final example, Example 6: Evaluate the limit as n goes to infinity of the sequence (n - 1)!/(n + 1)!.
00:31:11.100 --> 00:31:15.100
The first thing we want to do here is think, "Well, we don't really see how to do this immediately;
00:31:15.100 --> 00:31:18.700
so we want to see if we can simplify this into something where we have less going on."
00:31:18.700 --> 00:31:22.100
Factorials--it is kind of hard to see what is going on with factorials.
00:31:22.100 --> 00:31:26.400
So, maybe let's get a sense for if there is some way to cancel them and expand things.
00:31:26.400 --> 00:31:30.900
We realize that they are both based around a somewhat similar thing.
00:31:30.900 --> 00:31:37.200
n + 1 isn't very far from n - 1; so we can expand the factorials, so that we can cancel out based on that.
00:31:37.200 --> 00:31:46.200
We have (n - 1)! on top, and (n + 1)!...well, that is going to be n + 1 times one less than that, which is going to be n,
00:31:46.200 --> 00:31:49.200
times one less than that, n - 1, times one less than that...
00:31:49.200 --> 00:31:52.400
Well, if we keep going down forever, that is going to be (n - 1)! here.
00:31:52.400 --> 00:31:57.700
So, we have (n - 1)! on the top and n + 1 times n times (n - 1)! on the bottom.
00:31:57.700 --> 00:32:08.900
Well, we can cancel the (n - 1)!'s now; and we have the limit as n goes to infinity of 1/(n + 1)(n).
00:32:08.900 --> 00:32:15.100
So now, we can see that, as n goes off to infinity, well, our top doesn't change at all; it is just a constant in this case.
00:32:15.100 --> 00:32:20.500
So, since our top isn't ever going to change, but our bottom, (n + 1) times n, is going to get larger and larger and larger
00:32:20.500 --> 00:32:24.500
as n goes off to infinity, that means our bottom is growing, but our top is just staying the same.
00:32:24.500 --> 00:32:30.800
So, in the long run, it is going to get crushed down to 0; the fraction will get crushed down to 0, so the limit of this sequence is 0.
00:32:30.800 --> 00:32:34.800
All right, that finishes our exploration of limits in this course.
00:32:34.800 --> 00:32:38.200
We are now going to move on to derivatives, and we will get a cool sense for how derivatives work.
00:32:38.200 --> 00:32:42.300
It is really great stuff; we are getting a chance to see a preview of calculus, which is going to be really useful
00:32:42.300 --> 00:32:44.400
for when we get to calculus, because we are setting a groundwork here
00:32:44.400 --> 00:32:47.400
that you will then be able draw upon later, when you learn this stuff again.
00:32:47.400 --> 00:32:49.000
All right, we will see you at Educator.com later--goodbye!