In this lesson, our instructor Vincent Selhorst-Jones teaches Limits at Infinity & Limits of Sequences. Youll learn about limit of a function at infinity and how to evaluate them. Youll explore the three possibilities and two good ways to look at them. Next, Vincent goes over the limit of a sequence and numerical evaluation. The lesson ends with six fully worked-out practice examples.
Important Idea: infinity is not a location. It is the idea of going on forever, moving on to ever larger numbers. You can travel towards ∞, but you can never reach ∞.
We denote the limit of a function at infinity with
They mean the value that f(x) approaches as x goes off toward −∞ or ∞, respectively.
A limit at infinity works very similarly to how a normal limit works. Does the function "settle down" to a specific value L? It's just in this case, we're asking about the long-term behavior instead of x→ c. [It is important to note that most of
the functions we're used to working with do not have limits at infinity.]
The type of functions we will work with most often that can have a limit at infinity are rational functions. We worked with these many lessons ago when we learned about asymptotes. They are functions created from a fraction with polynomials in
the numerator and denominator:
an xn + an−1 xn−1 + …+ a1 x + a0
bm xm + bm−1 xm−1 + …+ b1 x + b0
Notice that n is the numerator's degree and m is the denominator's degree. Then we can find limits at infinity by the below:
If n < m ⇒ limx→ ±∞ f(x) = 0
If n=m ⇒ limx→ ±∞ f(x) = [(an)/(bm)]
If n > m ⇒ limx→ ±∞ f(x) does not exist
In general, when trying to find limits at infinity, think in terms of how the function will be affected as x grows very large (positive and negative). Does the function grow without bound? Will it "settle down" over time? Two good ways to think about
What happens if we plug in a LARGENUMBER?
What are the rates of growth in the function? Which parts grow faster? Which parts grow slower?
By thinking through these questions, you can get a good idea of how the function will behave over the long-term.
We can take this idea of a limit going to infinity and apply it to a sequence as well. Let a1, a2, a3,... be some infinite sequence. Then we can consider
The limit of a sequence is very similar to the limit of a function at infinity. Does the sequence "settle down" to a specific value L as n→ ∞?
Sometimes it can be difficult to tell how a function or sequence will behave in the long-run. In that case, we can evaluate the function numerically: plug in numbers and see what comes out. By using a calculator, we can plug in very large numbers
(positive and negative) and see what happens to the function or sequence. Doing this will give us a good sense of the long-term behavior. Similarly, if you have access to a graphing calculator or program, you can graph the function. Expand the viewing
window to a large horizontal region and look to see if the graph "settles down" in the long-run.
Limits at Infinity & Limits of Sequences
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.