# How do you solve the following limit #(e^x-1)/x# as x approaches zero?

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It is a remarkable limit, but, if you want to demonstrate it, you have to know the fundamental limit:

let

So:

let

and if

(for the second limit)

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The limit is in the indeterminate form

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#### Explanation:

We have,

The series expansion of

Hence,

#color(crimson)\Rightarrow = 1 +x/(2!) + x^2/(3!) + x^3/(4!) + \cdots#

Applying limit,

#lim_(x \to 0) f(x) = lim_(x \to 0) (1 +x/(2!) + x^2/(3!) + x^3/(4!) + \cdots) = 1#

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