Find $ \lim_{{x \to 1}} \frac{{xe^x - e}}{{x^2 - 1}} $. Choose 1 answer: A) $ \frac{e}{2} $ B) $ e $...

Question

Find $ \lim_{{x \to 1}} \frac{{xe^x - e}}{{x^2 - 1}} $. Choose 1 answer: A) $ \frac{e}{2} $ B) $ e $ C) $ \frac{1}{2} $ D) The limit doesn't exist.

SnapAndSolve AI · Accepted Answer

To solve this limit, we can use L'Hôpital's Rule because both the numerator and the denominator approach 0 as $ x \to 1 $.

First, differentiate the numerator and the denominator:

Now apply L'Hôpital's Rule:

$ \lim_{{x \to 1}} \frac{{xe^x - e}}{{x^2 - 1}} = \lim_{{x \to 1}} \frac{{e^x + xe^x}}{{2x}} $

Substitute $ x = 1 $:

$ \frac{{e^1 + 1 \cdot e^1}}{{2 \cdot 1}} = \frac{{2e}}{2} = e $

Answer: B) $ e $

Find \( \lim_{{x \to 1}} \frac{{xe^x - e}}{{x^2 - 1}} \). Choose 1 answer: A) \( \frac{e}{2} \) B) \( e \)...