Evaluate lim(x→0) (a^x - b^x)(x)

Question

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

lim(x→0) (a^x - b^x)/(x)

Answer 1

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

lim(x→0) (a^x - b^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 \(\lim\limits_{\text x \to0}\cfrac {a^{\text x}-b^{\text x}}{\text x}\) is In \(\cfrac ab\).