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/4 cot x−1 / x−π/4

Accepted Answer

Hello. In this video, we are going to be determining the value of the given limit. So we're asked to find the value of the limits. As X approaches pi divided by 2 of the following function. Sign of X minus 1, divided by the quantity, X minus pi divided by 2. So, let's go ahead and plug in the limit to see what the value of the limit is going to be. So we will directly substitute the value of the limit, which is pi divided by 2 for every variable X within our function. That is going to give us a sign of pi divided by 2, minus 1 divided by the quantity, pi divided by 2 minus pi divided by 2. Now, at this point, there is going to be an issue when we plug in the value with pi divided by 2. Sign of pi divided by 2 is going to simplify to be 1, and 1 minus 1 will give us 0. Furthermore, pi divided by 2 minus pi divided by 2 will produce 0. So what we have is 0 divided by 0, and this is what we call an indeterminate form. So what we have is a limit in an indeterminate form. So how can we fix this limit in a way that will allow us to find the overall value? Well, since we are dealing with an indeterminate form, we are going to use Lahopito's law. Now the Hopito's law tells us that if we have a rational function in the form of an indeterminate form, then what we want to do is you want to take the individual derivatives of both the numerator and denominator. After taking the individual derivatives, we will then reapply the value of the limit. So let's go ahead and start by taking the derivative of our numerator, sin of x minus 1. Now, by definition, the derivative of sin of X will be positive cosine of X, and the derivative of -1 will be 0, because the derivative of a constant is 0. In the denominator, the derivative of X is going to be 1 by the power rule, and the derivative of negative pi divided by 2 will be 0, since it is a constant value. This is going to simplify to be cosine of X divided by 1, which will just be cosine of X. Now let's go ahead and apply the limit, as X approaches pi divided by 2. When we apply the limit, we get cosine of pi divided by 2, which is equal to 0. And this is going to be the value of the limit by using Lehopito's law, and the overall solution to the problem is going to be B. So I hope this video helps you in understanding how to use the Hopitos law, and I will go ahead and see you all in the next video.

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/4 cot x−1 / x−π/4

Key Concepts

Limits

Derivatives

Cotangent Function

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