limit x->infinity x/e^x — step-by-step solution

Let's evaluate the limit of

\frac{x}{e^{x}}

when

x \to +∞

The limit of

e^{x}

+∞

when

x \to +∞

For

\frac{x}{e^{x}}

, we encounter the indeterminate form

\frac{+∞}{+∞}

x

tends to

+∞

Let's apply L'Hôpital's rule (limit of a/b = limit of a'/b') by differentiating numerator and denominator:

Let's perform the differentiation:

{(x)}^{'}

1

Let's perform the differentiation:

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

e^{x}

Using those derivatives, we obtain a new fraction whose limit can be computed:

The limit of

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

0

⁺ when

x \to +∞

So the limit of

\frac{x}{e^{x}}

0

⁺ when

x \to +∞

▷Answer: $0$