limit x->infinity (exp(x)-1)/(exp(x)-x)

1) Compute the limit of

\frac{\exp (x) - 1}{\exp (x) - x}

x \to +∞

ⓘ This is a reference limit of the function exp

The limit of

\exp (x)

+∞

when

x \to +∞

ⓘ A quantity that grows without bound plus a finite or bounded quantity still grows without bound

The limit of

\exp (x) - 1

+∞

when

x \to +∞

For

\exp (x) - x

, we obtain the indeterminate form

+∞ + -∞

x

tends to

+∞

By comparative growth,

\exp (x)

grows faster than

x

x \to +∞

The limit of

\exp (x) - x

+∞

when

x \to +∞

For

\frac{\exp (x) - 1}{\exp (x) - x}

, we encounter the indeterminate form

\frac{+∞}{+∞}

x

tends to

+∞

2) To determine the limit, divide the numerator and denominator by

\exp (x)

3) Let's reduce:

\frac{\exp (x) - 1}{\exp (x)}

ⅰ

\frac{\exp (x) - 1}{\exp (x)}

1 - \frac{1}{\exp (x)}

1 - \frac{1}{\exp (x)}

4) Let's simplify:

\frac{\exp (x) - x}{\exp (x)}

ⅰ

\frac{\exp (x) - x}{\exp (x)}

1 - \frac{x}{\exp (x)}

1 - \frac{x}{\exp (x)}

We obtain a new fraction whose limit can be determined:

\frac{1 - \frac{1}{\exp (x)}}{1 - \frac{x}{\exp (x)}}

ⓘ A number divided by a quantity that grows without bound tends to 0

The limit of

- \frac{1}{\exp (x)}

0

⁻ when

x \to +∞

ⓘ The limit of a sum of finite limits is the sum of the limits

The limit of

1 - \frac{1}{\exp (x)}

1

when

x \to +∞

For

- \frac{x}{\exp (x)}

, we encounter the indeterminate form

\frac{+∞}{+∞}

x

tends to

+∞

By comparative growth,

\exp (x)

grows faster than

x

x \to +∞

The limit of

- \frac{x}{\exp (x)}

0

⁻ when

x \to +∞

ⓘ The limit of a sum of finite limits is the sum of the limits

The limit of

1 - \frac{x}{\exp (x)}

1

when

x \to +∞

The limit of

\frac{1 - \frac{1}{\exp (x)}}{1 - \frac{x}{\exp (x)}}

1

when

x \to +∞

So the limit of

\frac{\exp (x) - 1}{\exp (x) - x}

1

when

x \to +∞

▶Limit:

1