Determine the following limits.
lim t→∞ (5t^2

Question

Determine the following limits.

lim t→∞ (5t² + t sin t) / t²

Accepted Answer

Welcome back, everyone. In this problem, we want to find the limit as X approaches infinity of 11 X to the 5th plus X the 4th plus X, all divided by X to the 5th. A says it's 12, B 0, C infinity, and D says it's 11. Now, if we're going to figure out this limit, let's first start by trying to simplify our function, OK? No. Simplifying 11 X to the 5th plus, OK. X to the 4th cost X all divided by X to the 5th. Means we're going to divide each term by 15th. OK. I know when we do that, 11 X 5 divided by X 5 equals 11, and X 4 x divided by X 5 equals x divided by X. OK? So that's what our expression can look like. But here, if we're going to solve the limit, OK, then we'll need to find the limit as X approaches infinity of 11. And as X approaches infinity of cast X divided by X. Now we know 11 is a constant, so that limit will be 11. We'll come back to that in a minute. But for now, let's focus on cast X divided by X. What can help us to figure out our limit here? Well, first, if we not, OK. That class X is between -1. And positive one inclusive, OK. Then if we divide our expression by X. Then we'll get -1 divided by X is less than or equal to x divided by X, which is less than or equal to 1 divided by X. Now why did we write this? Well, notice that we have a boundary here and we should be able to find the limits of both terms and the end of our boundary. Now recall that by the squeeze theorem. OK. At the squeeze theorem. OK, if FFX. Is less than or equal to G of X, which is less than or equal to HFX. In other words, if GFX is squeezed between FFX and HFX, yeah. And sorry, when X. Is near a. Except possibly at a, OK. Except possibly at it. And the limit. Of as X approaches A of FX is the same as the limit as X approaches A of G of X, OK, which is some, let's call this value L. then. Sorry, this should be H of X. My apologies. The limit as X approaches A H X equals L. Then the limit of G of X as X approaches A is also going to be equal to L. Now this is a very important outcome. Because if we can find the limits of both these terms, -1 divided by X and 1 divided by X, it will be the same limit for the cosine of X divided by X. Now, what are both of those limits? Well, starting with our function on the left, starting with F of X, OK. If we look at the limit as X approaches infinity. Of -1 divided by X, then as x approaches infinity, the denominators of -1 divided by X becomes arbitrarily large. Thus, the limit of -1 divided by X as X approaches infinity will be 0. And the same thing will happen with the limit as x approaches infinity of 1 divided by X. It will also be equal to 0. Now since we have both our terms here equal to 0, OK, that means the limit of the function squeezed between them, the cosine of x divided by X as X approaches infinity, is also going to be equal to 0. So if we know that, no, we can see. That the limit as X approaches infinity of our function, which we can rewrite as a level plus the cosine of X divided by X is going to be equal to, let's put that in a bracket here. That's gonna be equal to the limit as X approaches infinity of 11. Plus the limit as x approaches infinity of the cosine of x divided by X, and we know the limit of 11 as X approaches infinity equals 11, and as we've already established the limit of the cosine of x divided by X as X approaches infinity is 0. 11 + 0 equals 11. Thus our limit at infinity is 11. If we look back at our answer choices, that tells us B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Determine the following limits.
lim t→∞ (5t² + t sin t) / t²

Key Concepts

Limits at Infinity

Dominant Terms

L'Hôpital's Rule

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