limit x->infinity x^2/e^x — step-by-step solution

1) Let's evaluate the limit of

\frac{x^{2}}{e^{x}}

when

x \to +∞

The limit of

x^{2}

+∞

when

x \to +∞

The limit of

e^{x}

+∞

when

x \to +∞

For

\frac{x^{2}}{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:

Let's differentiate:

{(x^{2})}^{'}

2 x

3) Let's differentiate:

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

e^{x}

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

The limit of

2 x

+∞

when

x \to +∞

For

\frac{2 x}{e^{x}}

, the limit takes the indeterminate form

\frac{+∞}{+∞}

when

x

→

+∞

4) 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:

{(2 x)}^{'}

2

5) Let's differentiate:

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

e^{x}

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

The limit of

\frac{2}{e^{x}}

0

⁺ when

x \to +∞

The limit of

\frac{2 x}{e^{x}}

0

⁺ when

x \to +∞

So the limit of

\frac{x^{2}}{e^{x}}

0

⁺ when

x \to +∞

▷Result: $0$