# Limit of (a^x-1)/x

In this tutorial we shall discuss another very important formula of limits,

Let us consider the relation

Let $y = {a^x} - 1$, then $1 + y = {a^x}$, we have

Consider the relation

Using the logarithm on both sides, we have

Also $\mathop {\lim }\limits_{x \to 0} y = \mathop {\lim }\limits_{x \to 0} \left( {{a^x} - 1} \right) = {a^0} - 1 = 1 - 1 = 0$.

This shows that $y \to 0$ as $x \to 0$. Therefore, the given limit can be written as

Using the relation $\mathop {\lim }\limits_{x \to 0} {\left( {1 + x} \right)^{\frac{1}{x}}} = e$, we have

But $\ln e = 1$, we have