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

1) Compute the limit of

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

x \to +∞

ⓘ A number greater than 1 raised to a power that grows without bound grows without bound, a number between 0 and 1 tends to 0

The limit of

e^{x}

+∞

when

x \to +∞

For

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

, we encounter the indeterminate form

\frac{+∞}{+∞}

x

tends to

+∞

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

3) Let's differentiate:

\frac{d}{d x} (x)

ⓘ The derivative of x is 1:

\frac{d}{d x} (x)

1

1

4) Let's perform the differentiation:

\frac{d}{d x} (e^{x})

ⓘ The derivative of exp(x) is exp(x):

\frac{d}{d x} (e^{x})

e^{x}

e^{x}

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

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

The limit of

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

0

⁺ when

x \to +∞

So the limit of

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

0

⁺ when

x \to +∞

▶Limit:

0