WEBVTT mathematics/ap-calculus-ab/hovasapian
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Hello, welcome back to www.educator.com, and welcome back to AP Calculus.
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Today, we are going to do some example problems for the limit of a function.
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Let us jump right on in.
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The first one says, explain in words what the following symbols mean.
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The limit as x approaches 6 of f(x) = 14, what does this mean?
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It means the following.
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I think I will work in red here.
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This one says, as x gets arbitrarily close to 1,
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I do not like arbitrarily close, let us just say closer and closer indefinitely.
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As x gets closer and closer to 6 from below and above,
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because we do not have that positive or negative superscript on the number, f(x) gets closer and closer to 14.
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That is all this symbol says.
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As x gets arbitrarily close to 6 from below and from above, f(x) is getting close to 14 from below and from above.
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How about this one, the limit as x approaches 4 from below of f(x) = infinity?
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This one says, as x gets close to 4 from below, from the left, it is going to be interchangeable.
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Sometimes, we will say from below.
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Sometimes, we will say from the left.
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Sometimes, we will just say left hand limit, things like that.
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They are all synonymous.
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As x gets close to 4 from the left, f(x) goes off to positive infinity.
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This one, the limit as x approaches 4 from the right is equal to -3.
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This one says, as x approaches 4 from the right, this time I will go ahead and write from above,
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all of these things is synonymous, f(x) approaches the number -3.
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It gets closer and closer to, approaches, all of these words are going to be used.
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Given what you have just written, for the first limit, this one, can f(x) equal 5?
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The limit as x approaches 6 of f(x) is 14, can f(x) = 5?
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Absolutely, yes, it is not a problem at all because we know that the limit of a function,
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as it gets close to a number is independent of the value of the function at the number.
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Yes, not a problem of all.
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The limit as x approaches 6 of f(x).
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It is starting to get a little crazy now with my writing.
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Is independent of what happens when x = 6.
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Draw all possible graph for the second limit above.
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The second limit above that is that one right there.
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The limit as x approaches 4 from below, this is not as x approaches -4.
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As x approaches 4 from below, from the left of f(x) goes off to infinity.
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We have something like this.
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Let us say this is 4, we are going to approach 4 from below which means we are going to approach it this way.
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We are going to take x value getting closer and closer, and then, it is going to go off to positive infinity.
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This positive infinity, this is an asymptote.
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This might be a graph right there.
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As it gets closer and closer to 4, the function blows up to infinity.
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That is it, very straightforward, just go with your intuition.
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Clarify the following limit if it exists.
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The limit as x approaches 0 of sin x/ x.
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As of right now, you have two intuitive tools that you can use, in order to find the limit.
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You have the graph of the function and you can make a table of values.
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Taking values closer and closer to whatever number that you are approaching.
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You can use both, and both are very powerful tools.
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Clearly, the tabular gives you a better idea of what is happening because it gives you specific numbers.
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But the graph is perfect because it gives you a nice first approximation for what is happening.
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In this particular case, as x approaches 0, notice we are not specifying left hand or right hand limit.
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We have to do both.
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We have to approach 0 from the left from below and we have to approach it from the right from above.
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In each case, when we look at the graph, the function looks like it is approaching 1.
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And then from the right, the function is climbing and climbing, it looks like it approaches 1.
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Let us go ahead and confirm that with our table of values.
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From here to here, this is x approaching 0 from below -0.5, -0.25, 0.1, -0.01, -0.001.
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We see it is going from 0.95, 0.9999, definitely it looks like it is getting close to 1.
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Here x is approaching 0 from above 0.5, 0.25, 0.1, 0.01, 0.001.
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We are getting closer and closer and closer to 0.
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We see that the value of the function itself is going 0.95, 0.98, 0.998, 0.99999999999.
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Yes, we can conclude that this limit is equal to 1.
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You use your graph, you use your table of values.
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In a subsequent lesson, we are going to learn how to do this analytically.
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Example 3, use the graph below of f(x) to find the following limits, if they exist.
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If they do not exist, give the reason why.
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This is our function right here, we see it down below.
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We have solid dot here.
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Any place where on these graphs where you actually do not see an open dot or open, this is an open circle here.
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It is going to be an open circle here.
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Let me actually work in blue.
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We see that at 3, the value of the function has actually defined what is that solid dot.
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It is over here at 2.5.
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The limit as x approaches 1.5 of f(x), 1.5 is here.
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It does not specify, we have to approach it this way and approach it this way.
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What happens to the function?
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The function looks like it is getting close to 4.
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From here, it looks like f is getting close to 4.
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This limit is equal to 4, nice and simple, very straightforward.
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The e to the left and right hand limits equal each other.
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The limit exist and the limit = 4.
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The limit as x approaches 3.
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The limit as x approaches 3 from here, let us actually do c and d first because here it says 3 from below and 3 from above.
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These two together are the same as what b is.
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The limit as x approaches 3 from below, following this way, the y values looks like it is going to be right about there.
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That looks like it is going to be about 3. 15, something like that.
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Let us say this approximately equals like 3.2, and then, the limit as x approaches 3 from above.
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When we come this way, as we get close to 3 from this way, the function is getting close to 1.
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This one is equal to 1.
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The left hand limit 3.20, right hand limit 1.
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Therefore, this limit does not exist.
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The left hand limit exists, the right hand limit exists, but because these are not equal, the limit itself does not exist at 3.
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The limit as x approaches -2 from above.
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Notice, this negative in front, that is the number.
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The positive, that is from above.
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We are approaching -2 from above.
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What is the function doing, it looks like it is getting close to that number right there.
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It looks like that number is about, let us say 2.1.
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What is f(3)?
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The value of f(3) is 2.5.
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Notice the left hand limit is about 3.2.
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The right hand limit, the limit from above is 1.
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The value of the function at 3 is 2.5.
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These are all different, they are independent.
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You can have a left hand limit, you can have a right hand limit.
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You are going to have the value of the function at that point.
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They do not have to be equal.
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They are incompletely independent.
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If the two limits are equal, we say the limit exists.
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If the two limits are equal and it happens that equal the value of the function, then we say that the function is continuous.
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That is a stronger property.
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Here, clearly, the function is discontinuous.
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Because you see that I’m drawing my curve, I have to lift my pencil and that continue on.
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This is a discontinuity.
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If all three are equal which we will get to, we actually call that continuous.
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That is a very special property, when all three are equal.
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Let us go ahead and try another one.
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Use the graph below of f(x) to find the following limit, if they exist.
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Same thing that we just did, if they do not exist, give the reason why.
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The limit as x approaches a -1.
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-1 is here, x approaches -1, it does not specify.
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We have to do a left hand limit and we have to do a right hand limit.
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The left hand limit, as x approaches, we see that the function approaches this value right there, whatever that happens to be.
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The right hand limit as we approach -1 from the right, the function is this.
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The limit is here.
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Clearly, this number and this number are not the same.
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The limit as x approaches negative of f(1), it does not exist.
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What is the value of f(-1), it is this one right here.
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It is actually equal to 1.
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What is the limit as x approaches 1 from below?
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1 from below, the limit is 1.
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The limit as x approaches 1 from above.
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From above, the function is approaching -1.
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A left hand limit exists, a right hand limit exists.
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The limit itself does not exist.
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What is the value of f(1)?
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It turns out that the left hand limit corresponds with the value of the function at the point.
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Notice, this is equal to that but it is not equal to this.
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This is not continuous.
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You can see from the graph that it is not continuous.
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If you did not have a graph, if you just had this and this, you could say that is not continuous.
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You do not need the graph, the limits will tell you whether something is continuous or not.
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The limit as x approaches 3 of f(x).
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3 is over here, we are going to approach it from below.
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As we approach it from below, it looks like the graph is approaching -1.
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As we approach it from above, it approach the value 3 from above, it is x does this.
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The value of the function itself approaches -1.
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Here, the left hand limit which is -1, right hand limit -1.
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They equal, the limit = -1.
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In this case, the value of the function at 3, in other words, f(3) also happens to equal -1.
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When all three are equal, this is a continuous function.
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As you can see, this is a nice continuous function.
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It is perfectly smooth, in other words.
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When we say it is continuous, we mean smooth.
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There is no break, I do not have to lift my pencil to continue the graph.
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This is a discontinuous function, there is a discontinuity at -1 and there is a discontinuity at +1.
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There are left hand and right hand limits, they do not equal each other.
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A left hand and right hand limit that do not equal each other.
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At 3, it is continuous.
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The left hand limit, the right hand limit, and the value of the function equal each other.
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Sketch the graph of the function that satisfies the following properties.
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Very simple, you just have to make a graph that satisfies these properties.
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The limit as x approaches 0 from below is negative infinity.
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The limit as x approaches 0 from above is positive infinity.
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I know the graphs goes, let me do the graph in red actually.
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Negative infinity, this one, as I approach 0 from below, my graph goes to negative infinity.
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As I approach 0 from above, it goes to positive infinity.
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The limit as x goes to negative infinity, as x gets really big in the negative direction is 0.
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I know that it gets close to here, maybe not.
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The limit as x approaches 2 of f(x) is equal to 3.
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The limit as x approaches positive infinity is equal to 1.
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Let us deal with this one, as x approaches positive infinity.
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Let us stay over here on the right side of the graph, +1.
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I’m going to go ahead and draw an asymptote at 1.
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It tells me that as x goes to positive infinity, f(x) gets close to 1.
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I’m going to go ahead and draw that.
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Here as x approaches negative infinity, f(x) goes to 0.
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I know I'm here, this is going to go probably, it is a possibility.
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It is not the only possibility but this is the possibility.
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f(2) is undefined.
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As x approaches 2, it equal 3.
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Let us say this is 2, let us say this is 3.
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I have to actually connect it.
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It is undefined there, there is going to be a hole there.
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But the limit as x approaches 2, meaning from the left and from the right, the limit is the same.
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The function actually touches here and goes down that way.
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There we go, that is a much better function.
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That satisfies all of the properties.
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f(2) is undefined.
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As x approaches 0 from below, the graph goes to negative infinity.
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As x approaches 0 from above, the graph goes to positive infinity.
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As x goes to negative infinity, the graph goes to 0.
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Yes, the y value goes to 0.
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As x approaches 2 from below, from above the graph, it approaches the number 3.
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That takes care of that.
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As x goes to infinity, f(x) approaches the value 1.
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Yes, that is the horizontal asymptote and f(2) is undefined.
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This is a graph, not the only graph, there might be others but that is a good graph.
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Find the following limit if it exists.
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The limit as x approaches 0 of this function.
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Here, we have the graph of the function.
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We are going to use our graph to let us know what is happening as x approaches 0.
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X approaches 0, 0 is a number.
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It does not specify whether it is left or right hand limit, we have to do both.
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We have to approach 0 from below, 0 from above.
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As we approach 0 from below, the limit of f(x) as x approaches 0 from below.
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As we approach it from below, the function, the y value goes to positive infinity.
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The limit as x approaches 0 from above of this f(x).
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As we go this way, the function goes to negative infinity.
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The limit does not exist.
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That is it, nice and simple.
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Just use the graph to tell you what is going on.
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If you need a table of values, you can use a table of values.
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But here, it is clear, you can see what the graph is doing.
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They are blowing up to infinity but in opposite directions.
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If this 1 to infinity from the left and from the right, we can say the limit = positive infinity.
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Determine the following limit.
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The limit as x approaches 5 of sin x × ln x.
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Here we avail ourselves of both the graph and the table of values.
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From the graph, we are approaching 5.
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This is the number we are approaching.
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We are approaching it from this point and we are approaching if from the right.
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From below, from above.
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Left hand limit, right hand limit.
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The left hand limit, as it approaches this number, we see that it gets close to some value.
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As we approach x, we approach 5 from the right.
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We see that the y value approaches the same value.
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Let us see what it is at, here, x approaches 5 from below.
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Here this is x approaching 5 from above.
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Let us see what the numbers do.
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4.9, 4.99, 4.999, 4.9999, getting really close to 5.
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We notice that we are approaching -1.5433.
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Here, same thing, we go to 5.1, 5.01, 5.001, 5.0001, getting closer and closer.
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We see that we are approaching the same number 1.5433, -1.5433.
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I can conclude, because the left hand limit and the right hand limit are the same,
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they look like they are the same number.
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Clearly, they are the same in the graph.
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The limit as x approaches 5 of the sin(x) × natlog(x).
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I can say that it equal -1.543.
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It is good to 3 decimal places.
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That is it, very intuitive notion actually.
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Yet, the entire modern world is based on this notion.
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The idea of the limit, the idea of the derivative.
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It is quite extraordinary.
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Let us go ahead, given f(x), let us round out what we have done here today.
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Given f(x), use your calculator or a graphing utility
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or graphing software to make a graph of the function that you are dealing with.
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Make a table of values and speculate about whether the limit converges, whether the function converges to a specific number.
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Use your graph, use your table of values and make some good choices.
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In the next lesson, we learn how to evaluate limits analytically.
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In other words, doing something mathematical.
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It is going to turn out that the math is really simple.
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Thank you for joining us here at www.educator.com.
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We will see you next time, bye.