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

1) Let's evaluate the limit of

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

when

x \to +∞

The limit of

\exp (x)

+∞

when

x \to +∞

The limit of

\exp (x) - 1

+∞

when

x \to +∞

For

\exp (x) - x

, the limit takes the indeterminate form

+∞ + -∞

when

x

→

+∞

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}

, the limit takes the indeterminate form

\frac{+∞}{+∞}

when

x

→

+∞

2) To find the limit, we divide the numerator and the denominator by

\exp (x)

Let's reduce:

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

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

3) Let's reduce:

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

The limit of

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

0

⁻ when

x \to +∞

The limit of

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

1

when

x \to +∞

For

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

, the limit takes the indeterminate form

\frac{+∞}{+∞}

when

x

→

+∞

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

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 +∞

▷Answer: $1$