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.

Find a possible function f and number 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. Limit as x approaches 0 of eats the power of 2 x minus 1 divided by X. We're given 4 answer choices A says G of X equals e to the power of 2 X and A is 1. B G X equals eats the power of 2 x and a is 0 C G X equals eats the power of x and a is 1 and D G X equals eats the power of x and a is 0. So if we are given a function G of X, it's derivative at a specific point, let's suppose G at a. This is what we're looking for. Using the limit definition, we can see that this can be written as limit. As x approaches A of G of X minus G of A. Divided by x minus a. And what we're going to do is simply use the analogy. Let's understand that we are given the limit as x approaches 0 of e to the power of 2 x minus 1 divided by X. Now what we notice is that our numerator simply contains a negative sign, and in our expression we only have one negative sign. So by analogy we can try and guess that X G of X is equal to e to the power of 2 X, which gives us our second term being G of A. So G A. Must be equal to one. Now if we consider our denominator, it also has a difference between X and A, but we're only given X. So what we can do is simply subtract 0 because X minus 0 is X, right? So if this is true, then X is simply X by analogy, and a must be 0. What we're going to do from here is simply check if our given function matches a value of 1 and 0, right, because we understand that GA must be 1. So if we take our proposed function G of X and if we evaluate it at 00 G.0, we're going to take E to the power of 2 multiplied by 0, which is E to the power of 0, and this is indeed a 1. So it matches 1, which means that our proposed function and the value of A is sensible. So we can conclude that G of X is eats the power of 2 X. We can exclude options C and D, and A is 0, right? So this means that the correct answer to this problem corresponds to the answer choice B. Thank you for watching.

The following limits represent f'(a) for some function f and some real number a.
Find a possible function f and number a.
lim x🠂0 e^x-1 / x

Key Concepts

Limits

Derivatives

Exponential Functions

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