Calculus Examples

lim x \to \infty \frac{x^{2}}{e^{x}}

$lim_{x \to \infty} \frac{x^{2}}{e^{x}}$

Step 1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1

Take the limit of the numerator and the limit of the denominator.

\frac{lim x \to \infty x^{2}}{lim x \to \infty e^{x}}

$\frac{lim_{x \to \infty} x^{2}}{lim_{x \to \infty} e^{x}}$

Step 1.2

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

\frac{\infty}{lim x \to \infty e^{x}}

$\frac{\infty}{lim_{x \to \infty} e^{x}}$

Step 1.3

Since the exponent

x

$x$ approaches

\infty

$\infty$ , the quantity

e^{x}

$e^{x}$ approaches

\infty

$\infty$ .

\frac{\infty}{\infty}

$\frac{\infty}{\infty}$

Step 1.4

Infinity divided by infinity is undefined.

Undefined

\frac{\infty}{\infty}

$\frac{\infty}{\infty}$

Step 2

Since

\frac{\infty}{\infty}

$\frac{\infty}{\infty}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

lim x \to \infty \frac{x^{2}}{e^{x}} = lim x \to \infty \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [e^{x}]}

$lim_{x \to \infty} \frac{x^{2}}{e^{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [e^{x}]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

lim x \to \infty \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [e^{x}]}

$lim_{x \to \infty} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [e^{x}]}$

Step 3.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 2

$n = 2$ .

lim x \to \infty \frac{2 x}{\frac{d}{d x} [e^{x}]}

$lim_{x \to \infty} \frac{2 x}{\frac{d}{d x} [e^{x}]}$

Step 3.3

Differentiate using the Exponential Rule which states that

\frac{d}{d x} [a^{x}]

$\frac{d}{d x} [a^{x}]$ is

a^{x} ln (a)

$a^{x} \ln (a)$ where

a

$a$ =

e

$e$ .

lim x \to \infty \frac{2 x}{e^{x}}

$lim_{x \to \infty} \frac{2 x}{e^{x}}$

lim x \to \infty \frac{2 x}{e^{x}}

$lim_{x \to \infty} \frac{2 x}{e^{x}}$

Step 4

Move the term

2

$2$ outside of the limit because it is constant with respect to

x

$x$ .

2 lim x \to \infty \frac{x}{e^{x}}

$2 lim_{x \to \infty} \frac{x}{e^{x}}$

Step 5

Apply L'Hospital's rule.

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Step 5.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 5.1.1

Take the limit of the numerator and the limit of the denominator.

2 \frac{lim x \to \infty x}{lim x \to \infty e^{x}}

$2 \frac{lim_{x \to \infty} x}{lim_{x \to \infty} e^{x}}$

Step 5.1.2

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

2 \frac{\infty}{lim x \to \infty e^{x}}

$2 \frac{\infty}{lim_{x \to \infty} e^{x}}$

Step 5.1.3

Since the exponent

x

$x$ approaches

\infty

$\infty$ , the quantity

e^{x}

$e^{x}$ approaches

\infty

$\infty$ .

2 \frac{\infty}{\infty}

$2 \frac{\infty}{\infty}$

Step 5.1.4

Infinity divided by infinity is undefined.

Undefined

2 \frac{\infty}{\infty}

$2 \frac{\infty}{\infty}$

Step 5.2

Since

\frac{\infty}{\infty}

lim x \to \infty \frac{x}{e^{x}} = lim x \to \infty \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [e^{x}]}

$lim_{x \to \infty} \frac{x}{e^{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [e^{x}]}$

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

2 lim x \to \infty \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [e^{x}]}

$2 lim_{x \to \infty} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [e^{x}]}$

Step 5.3.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$2 lim_{x \to \infty} \frac{1}{\frac{d}{d x} [e^{x}]}$

Step 5.3.3

Differentiate using the Exponential Rule which states that

$\frac{d}{d x} [a^{x}]$ is

$a^{x} \ln (a)$ where

$a$ =

$e$ .

$2 lim_{x \to \infty} \frac{1}{e^{x}}$

Step 6

Since its numerator approaches a real number while its denominator is unbounded, the fraction

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

$0$ .

$2 \cdot 0$

Step 7

Multiply

$2$ by

$0$ .

$0$

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