limit x->0 (e^x-1)/x — step-by-step solution

Let's evaluate the limit of

\frac{e^{x} - 1}{x}

when

x \to 0

The limit of

e^{x}

1

when

x \to 0

The limit of

e^{x} - 1

0

when

x \to 0

For

\frac{e^{x} - 1}{x}

, we encounter the indeterminate form 0/0 as

x

tends to

0

We can use L'Hôpital's rule (limif of a/b = limit of a'/b') by computing the derivatives of the numerator and the denominator:

Let's perform the differentiation:

{(e^{x} - 1)}^{'}

{(e^{x})}^{'} + 0

e^{x}

Let's perform the differentiation:

{(x)}^{'}

1

With the derivatives of the numerator and the denominator, we obtain a new fraction whose limit can be determined:

So the limit of

\frac{e^{x} - 1}{x}

1

when

x \to 0

▷Result: $1$