The following limits represent f'(a) for some function f ...

Question

The following limits represent f'(a) for some function f and some real number a.

b. Evaluate the limit by computing f'(a).

lim x🠂1 x¹⁰⁰-1 / x-1

Accepted Answer

Hello. In this video, we are going to be identifying the function G and the value of A, such that the following limit represents G A. Then we will evaluate the limit by finding the value of G A. Now, in a previous video, we did complete this first part of the problem. We determined that G of X. Must equal to 2 x cubed and the value of a must equal to 2 by using the definition of the derivative G A. That definition was G A is equal to the limit as X approaches A of G X minus G of A. Divided by X minus 8. So, by using the definition, we determined the first two parts of this problem. But now what we need to do is we need to evaluate the limit by finding the value of G A. So, in order to find G A, We need to simplify the given limit, which is the limit as X approaches to. Of 2 X cubed minus 16 divided by X minus 2. So, let's go ahead and start by simplifying the limit. In order to simplify the limit, we will first need to simplify the numerator. Notice that both terms in the numerator have a multiple of 2. So, what we will start by doing is we will go ahead and factor out a 2 from the numerator. That will leave us with the limit as X approaches 2 of 2 multiplied by X cubed minus 8 divided by X minus 2. We can factor X cubed minus 8 by using the difference of cubed terms. That will allow us to rewrite the limit, as the limit as X approaches 2 of 2 multiplied by X minus 2 multiplied by the quantity X2 + 2 X + 4. Is all divided by the quantity X minus 2. Notice that we have a factor of X minus 2 in both the numerator and denominator. So what we can go ahead and do is eliminate these quantities from the limit. That will leave us with the limit as X approaches 2 of 2 multiplied by X2 + 2 X + 4. And by using the constant property of the limit, we can take the constant of 2 and move it into the front of the limit. That will allow us to rewrite G A S 2 multiplied by the limit as X approaches 2 of the function X2. Plus 2 X + 4. Now that our limit is in its simplest form, we can go ahead and evaluate the limit by directly substituting the value of the limit for every variable X. In the limit, since X approaches the value of 2, we can directly substitute every variable X with the value of 2. That will leave us with 2 multiplied by the quantity, 2 squared, plus 2 multiplied by 2 + 4. No, 2 squared will simplify to give us 4. 2 multiplied by 2 will give us 4, and 4 + 4 + 4 will leave us with 12. So that is going to leave us with 2 multiplied by 12, which is going to simplify to be 24. 24 is going to be the value of G A, and that will be the solution to this problem. So, I hope this video helps you in understanding how to evaluate this type of limits, and I'll go ahead and see you all in the next video.

The following limits represent f'(a) for some function f and some real number a.
b. Evaluate the limit by computing f'(a).
lim x🠂1 x¹⁰⁰-1 / x-1

Key Concepts

Limits

Derivatives

L'Hôpital's Rule

Watch next