Evaluate the following limits or state that they do not exist....

Question

Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)

lim x→π/2 cos x/x−π/2

Accepted Answer

Welcome back, everyone. Determine the limit of the function F of X equals sin x divided by X minus pi as X approaches pi. We're given 4 answer choices A says infinity, B1, C-1, and D does not exist. So let's define the limit. It says the limit as x approaches pi. Of of x divided by X minus pi. First of all, let's apply the right substitution. We're going to get sin of pi divided by pi minus pi. And this gives us a 0 divided by the 0, which is an indeterminate form, and we have to recall that this is one of the indeterminate forms where we can apply the Lapito's rule, right? If we have 0 divided by 0 or infinity divided by infinity, those forms allow us to apply the Lapiels rule. So what we want to do is simply rewrite the limit limit as X approaches 5. Of sin of x divided by x minus pi and we want to say that this limit is going to be equal to the limit as x approaches pi of the derivative of the top functions of x divided by the derivative of the bottom function x minus pi. And now if we evaluate each derivative, we're going to get limit as x approaches pi. The derivative of sine is cosine of x. And now for the bottom function, the derivative of X is 1, and the derivative of pi is 0. It is a constant the derivative of negative pi to be accurate, right? And what we want to do from here is attempt substitution once again. So we get cosign. Of pi divided by 1. Now cosine of pi is equal to -1, so we are dividing -1 by 1, and we get -1. This is now a finite number, meaning we have successfully evaluated the limit and the correct answer corresponds to the answer choice C. Thank you for watching.

Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)
lim x→π/2 cos x/x−π/2

Key Concepts

Limits

Derivatives

L'Hôpital's Rule

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