Calculus Examples

$lim_{x \to \infty} \frac{x^{3}}{e^{x}}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{x \to \infty} x^{3}}{lim_{x \to \infty} e^{x}}$

Step 1.1.2

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

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

Step 1.1.3

Since the exponent $x$ approaches $\infty$ , the quantity $e^{x}$ approaches $\infty$ .

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

Step 1.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 1.2

Since $\frac{\infty}{\infty}$ 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.

$lim_{x \to \infty} \frac{x^{3}}{e^{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [x^{3}]}{\frac{d}{d x} [e^{x}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [x^{3}]}{\frac{d}{d x} [e^{x}]}$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$lim_{x \to \infty} \frac{3 x^{2}}{\frac{d}{d x} [e^{x}]}$

Step 1.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$lim_{x \to \infty} \frac{3 x^{2}}{e^{x}}$

Step 2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$3 lim_{x \to \infty} \frac{x^{2}}{e^{x}}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

$3 \frac{lim_{x \to \infty} x^{2}}{lim_{x \to \infty} e^{x}}$

Step 3.1.2

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

$3 \frac{\infty}{lim_{x \to \infty} e^{x}}$

Step 3.1.3

Since the exponent $x$ approaches $\infty$ , the quantity $e^{x}$ approaches $\infty$ .

$3 \frac{\infty}{\infty}$

Step 3.1.4

Infinity divided by infinity is undefined.

Undefined

$3 \frac{\infty}{\infty}$

Step 3.2

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$3 lim_{x \to \infty} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [e^{x}]}$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$3 lim_{x \to \infty} \frac{2 x}{\frac{d}{d x} [e^{x}]}$

Step 3.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$3 lim_{x \to \infty} \frac{2 x}{e^{x}}$

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$3 \cdot 2 lim_{x \to \infty} \frac{x}{e^{x}}$

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.

$3 \cdot 2 \frac{lim_{x \to \infty} x}{lim_{x \to \infty} e^{x}}$

Step 5.1.2

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

$3 \cdot 2 \frac{\infty}{lim_{x \to \infty} e^{x}}$

Step 5.1.3

Since the exponent $x$ approaches $\infty$ , the quantity $e^{x}$ approaches $\infty$ .

$3 \cdot 2 \frac{\infty}{\infty}$

Step 5.1.4

Infinity divided by infinity is undefined.

Undefined

$3 \cdot 2 \frac{\infty}{\infty}$

Step 5.2

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

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

$3 \cdot 2 lim_{x \to \infty} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [e^{x}]}$

Step 5.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 \cdot 2 lim_{x \to \infty} \frac{1}{\frac{d}{d x} [e^{x}]}$

Step 5.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$3 \cdot 2 lim_{x \to \infty} \frac{1}{e^{x}}$

Step 6

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{e^{x}}$ approaches $0$ .

$3 \cdot 2 \cdot 0$

Step 7

Multiply $3 \cdot 2 \cdot 0$ .

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Step 7.1

Multiply $3$ by $2$ .

$6 \cdot 0$

Step 7.2

Multiply $6$ by $0$ .

$0$