First slide

Introduction to limits

Question

$lim_{x \to 0} \frac{a^{x} - b^{x}}{e^{x} - 1}$ is equal to

Moderate

A

$\log_{e} (\frac{a}{b})$
B

$\log_{e} (\frac{b}{a})$
C

$\log_{e} a b$
D

$\log_{c} (a + b)$

Solution

We have,
$\begin{array}{l} lim_{x \to 0} \frac{a^{x} - b^{x}}{e^{x} - 1} = lim_{x \to 0} \frac{(a^{x} - 1) - (b^{x} - 1)}{e^{x} - 1} \\ = lim_{x \to 0} \frac{(\frac{a^{x} - 1}{x}) - (\frac{b^{x} - 1}{x})}{(\frac{e^{x} - 1}{x})} = \frac{\log a - \log b}{1} = \log_{e} (\frac{a}{b}) \end{array}$

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