Find lim x→0 (1 - e^x) / ln(2 - e^x).
Choose 1 answer:
A) 2/e
B) 1
C) 0
D) The limit doesn't exist.

Question

Find lim x→0 (1 - e^x) / ln(2 - e^x). Choose 1 answer: A) 2/e B) 1 C) 0 D) The limit doesn't exist.

SnapAndSolve AI · Accepted Answer

To solve this limit, we can use L'Hôpital's Rule since both the numerator and the denominator approach 0 as x approaches 0.

Differentiate the numerator: d/dx(1 - e^x) = -e^x.

Differentiate the denominator: d/dx(ln(2 - e^x)) = -e^x / (2 - e^x).

Apply L'Hôpital's Rule:

lim x→0 (-e^x) / (-e^x / (2 - e^x)) = lim x→0 (2 - e^x).

As x approaches 0, e^x approaches 1, so the limit becomes 2 - 1 = 1.

The correct answer is B) 1.

Answer