Calculus Examples

$lim_{x \to \infty} \frac{e^{x}}{x^{4}}$

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.

$\frac{lim_{x \to \infty} e^{x}}{lim_{x \to \infty} x^{4}}$

Step 1.1.2

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

$\frac{\infty}{lim_{x \to \infty} x^{4}}$

Step 1.1.3

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

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

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.

$lim_{x \to \infty} \frac{e^{x}}{x^{4}} = lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x}]}{\frac{d}{d x} [x^{4}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x}]}{\frac{d}{d x} [x^{4}]}$

Step 1.3.2

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

$lim_{x \to \infty} \frac{e^{x}}{\frac{d}{d x} [x^{4}]}$

Step 1.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$lim_{x \to \infty} \frac{e^{x}}{4 x^{3}}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$\frac{lim_{x \to \infty} e^{x}}{lim_{x \to \infty} 4 x^{3}}$

Step 2.1.2

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

$\frac{\infty}{lim_{x \to \infty} 4 x^{3}}$

Step 2.1.3

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

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

Step 2.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 2.2

$lim_{x \to \infty} \frac{e^{x}}{4 x^{3}} = lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x}]}{\frac{d}{d x} [4 x^{3}]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x}]}{\frac{d}{d x} [4 x^{3}]}$

Step 2.3.2

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

$lim_{x \to \infty} \frac{e^{x}}{\frac{d}{d x} [4 x^{3}]}$

Step 2.3.3

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$lim_{x \to \infty} \frac{e^{x}}{4 \frac{d}{d x} [x^{3}]}$

Step 2.3.4

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

$lim_{x \to \infty} \frac{e^{x}}{4 (3 x^{2})}$

Step 2.3.5

Multiply $3$ by $4$ .

$lim_{x \to \infty} \frac{e^{x}}{12 x^{2}}$

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.

$\frac{lim_{x \to \infty} e^{x}}{lim_{x \to \infty} 12 x^{2}}$

Step 3.1.2

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

$\frac{\infty}{lim_{x \to \infty} 12 x^{2}}$

Step 3.1.3

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

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

Step 3.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 3.2

$lim_{x \to \infty} \frac{e^{x}}{12 x^{2}} = lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x}]}{\frac{d}{d x} [12 x^{2}]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x}]}{\frac{d}{d x} [12 x^{2}]}$

Step 3.3.2

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

$lim_{x \to \infty} \frac{e^{x}}{\frac{d}{d x} [12 x^{2}]}$

Step 3.3.3

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to \infty} \frac{e^{x}}{12 \frac{d}{d x} [x^{2}]}$

Step 3.3.4

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

$lim_{x \to \infty} \frac{e^{x}}{12 (2 x)}$

Step 3.3.5

Multiply $2$ by $12$ .

$lim_{x \to \infty} \frac{e^{x}}{24 x}$

Step 4

Apply L'Hospital's rule.

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Step 4.1

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

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Step 4.1.1

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

$\frac{lim_{x \to \infty} e^{x}}{lim_{x \to \infty} 24 x}$

Step 4.1.2

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

$\frac{\infty}{lim_{x \to \infty} 24 x}$

Step 4.1.3

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

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

Step 4.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 4.2

$lim_{x \to \infty} \frac{e^{x}}{24 x} = lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x}]}{\frac{d}{d x} [24 x]}$

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x}]}{\frac{d}{d x} [24 x]}$

Step 4.3.2

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

$lim_{x \to \infty} \frac{e^{x}}{\frac{d}{d x} [24 x]}$

Step 4.3.3

Since $24$ is constant with respect to $x$ , the derivative of $24 x$ with respect to $x$ is $24 \frac{d}{d x} [x]$ .

$lim_{x \to \infty} \frac{e^{x}}{24 \frac{d}{d x} [x]}$

Step 4.3.4

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

$lim_{x \to \infty} \frac{e^{x}}{24 \cdot 1}$

Step 4.3.5

Multiply $24$ by $1$ .

$lim_{x \to \infty} \frac{e^{x}}{24}$

Step 5

Since the function $e^{x}$ approaches $\infty$ , the positive constant $\frac{1}{24}$ times the function also approaches $\infty$ .

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Step 5.1

Consider the limit with the constant multiple $\frac{1}{24}$ removed.

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

Step 5.2

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

$\infty$