Evaluate lim(x→0) (a^x - a^-x)/x

Question

Evaluate \(\lim\limits_{\text x \to0}\left(\cfrac{a^{\text x}-a^{-\text x}}{\text x}\right) \)

lim(x→0) (a^x - a^-x)/x

Answer 1

To evaluate : \(\lim\limits_{\text x \to0}\left(\cfrac{a^{\text x}-a^{-\text x}}{\text x}\right) \)

lim(x→0) (a^x - a^-x)/x

Formula used: L'Hospital's rule

Let f(x) and g(x) be two functions which are differentiable on an open interval I except at a point a where

This represents an indeterminate form. Thus applying L'Hospital's rule, we get

Thus, the value of \(\lim\limits_{\text x \to0}\left(\cfrac{a^{\text x}-a^{-\text x}}{\text x}\right) \) is 2 In a.