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🠂0 e^x-1 / x

Accepted Answer

Welcome back, everyone. Identify the function G and value of A such that the following limit represents G at A. Then evaluate the limit by finding the value of G at A. We're given a limit as x approaches 0 of each to the power of 2 x minus 1 divided by X. Now let's simply understand that if we're given a function G of X, then we can evaluate G at any point A using the limit definition. It is going to be equal to limit as x approaches a of G X minus g of a divided by. X minus A. What we're going to do is simply equate this to the given limit and use the analogy. To make a good guess on what the function and the point A could be. So we're given limit as X approaches 0 of. E to the power of 2 X minus 1 divided by X. First of all, by analogy, we can see that x approaches A and on the other side x approaches 0. So immediately we can conclude that a is 0, and while in the denominator we have X minus A and X. And of course this is reasonable because X minus 0 is just X. So we have verified that A is indeed 0. Now if we consider our numerators, we have G of X minus g of A, and because in the given expression there's only one negative sign, it should be quite clear that G of X is most likely eats the power of 2 X. So let's assume that G of X is eats the power of 2 X, and then G of A must be 1. And we want to verify that, right? So G of 0, where our proposed function is going to be E to the power of 2 multiplied by 0, which is E to the power of 0, which is indeed 1. So we have Our A is equal to 0, and we also want to identify the function G. Our G of X is E to the power of 2 X. Our final step is to evaluate the limit by finding the value of G A. And for this purpose we're going to find the derivative of G of X. So we want to differentiate E to the power of 2 X, and we need to apply the chain rule. First of all, the derivative of E of X is just E to the power of X using the table of basic derivatives. In this case, our exponent is 2 X, so we're going to use 2 X. But because it is a composite function, we need to multiply that by the derivative of 2 X, which is simply 2, right? So we get G of X. Equals 2 e to the power of 2 X. And now we want to evaluate the G at point A and since A is 0. G at A is equal to 1 we already know that. I'm sorry, we know that GA is one and we want to evaluate G A. And essentially we can do that by taking G of X, which is 2 to the power of 2 X. But now instead of X, we are substituting the value of A. Which is 0, so we get 2 e to the power of 2 multiplied by 0, which is 2 multiplied by e to the power of 0, which is 2 multiplied by 1, and that's 2. So we can say that the final answer that we can include is the value of the limit. Limit. S X approaches 0 of each to the power of 2 X minus 1. Divided by x. Is simply the derivative of g at A, which is 0 right A is 0, and we have concluded that the derivative at point A is 2. So that'd be our final answers. We can label them and conclude the video. Thank you for watching.

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🠂0 e^x-1 / x

Key Concepts

Limits

Derivatives

Exponential Functions

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