Let $a 1$ and $b 1$ . Find the value of the limit $\lim _ { x \rightarrow 0 } \frac { a ^ { x } - b ^ { x } } { x }$ .
A)0
B)1
C) $a - b$
D) $b - a$
E) $\ln ( a - b )$
F) $\ln ( b - a )$
G) $\ln \left( \frac { a } { b } \right)$
H) $\ln \left( \frac { b } { a } \right)$

Question

Let $a > 1$ and $b > 1$ . Find the value of the limit $\lim _ { x \rightarrow 0 } \frac { a ^ { x } - b ^ { x } } { x }$ .
A)0
B)1
C) $a - b$ 
D) $b - a$ 
E) $\ln ( a - b )$
F) $\ln ( b - a )$
G) $\ln \left( \frac { a } { b } \right)$
H) $\ln \left( \frac { b } { a } \right)$

Quizplus · Accepted Answer

The limit can be solved by recognizing it as the derivative of $a^x - b^x$ at $x = 0$. The derivative of $a^x$ is $a^x \ln(a)$ and of $b^x$ is $b^x \ln(b)$, evaluated at $x = 0$ gives $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$.

Let a>1a > 1a>1 And b>1b > 1b>1 Find the Value of the Limit

Let $a > 1$ And $b > 1$ Find the Value of the Limit