Calculus Examples

$lim_{x \to \infty} x^{5} e^{- x^{4}}$

Step 1

Rewrite $x^{5} e^{- x^{4}}$ as $\frac{x^{5}}{e^{x^{4}}}$ .

$lim_{x \to \infty} \frac{x^{5}}{e^{x^{4}}}$

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.

$\frac{lim_{x \to \infty} x^{5}}{lim_{x \to \infty} e^{x^{4}}}$

Step 2.1.2

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

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

Step 2.1.3

Since the exponent $x^{4}$ approaches $\infty$ , the quantity $e^{x^{4}}$ approaches $\infty$ .

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

Step 2.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 2.2

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.

$lim_{x \to \infty} \frac{x^{5}}{e^{x^{4}}} = lim_{x \to \infty} \frac{\frac{d}{d x} [x^{5}]}{\frac{d}{d x} [e^{x^{4}}]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [x^{5}]}{\frac{d}{d x} [e^{x^{4}}]}$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$lim_{x \to \infty} \frac{5 x^{4}}{\frac{d}{d x} [e^{x^{4}}]}$

Step 2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{4}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u$ as $x^{4}$ .

$lim_{x \to \infty} \frac{5 x^{4}}{\frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{4}]}$

Step 2.3.3.2

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

$lim_{x \to \infty} \frac{5 x^{4}}{e^{u} \frac{d}{d x} [x^{4}]}$

Step 2.3.3.3

Replace all occurrences of $u$ with $x^{4}$ .

$lim_{x \to \infty} \frac{5 x^{4}}{e^{x^{4}} \frac{d}{d x} [x^{4}]}$

Step 2.3.4

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

$lim_{x \to \infty} \frac{5 x^{4}}{e^{x^{4}} (4 x^{3})}$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Reorder the factors of $e^{x^{4}} (4 x^{3})$ .

$lim_{x \to \infty} \frac{5 x^{4}}{4 e^{x^{4}} x^{3}}$

Step 2.3.5.2

Reorder factors in $4 e^{x^{4}} x^{3}$ .

$lim_{x \to \infty} \frac{5 x^{4}}{4 x^{3} e^{x^{4}}}$

Step 2.4

Cancel the common factor of $x^{4}$ and $x^{3}$ .

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Step 2.4.1

Factor $x^{3}$ out of $5 x^{4}$ .

$lim_{x \to \infty} \frac{x^{3} (5 x)}{4 x^{3} e^{x^{4}}}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $x^{3}$ out of $4 x^{3} e^{x^{4}}$ .

$lim_{x \to \infty} \frac{x^{3} (5 x)}{x^{3} (4 e^{x^{4}})}$

Step 2.4.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{x^{3} (5 x)}{x^{3} (4 e^{x^{4}})}$

Step 2.4.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{5 x}{4 e^{x^{4}}}$

Step 3

Move the term $\frac{5}{4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{5}{4} lim_{x \to \infty} \frac{x}{e^{x^{4}}}$

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.

$\frac{5}{4} \cdot \frac{lim_{x \to \infty} x}{lim_{x \to \infty} e^{x^{4}}}$

Step 4.1.2

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

$\frac{5}{4} \cdot \frac{\infty}{lim_{x \to \infty} e^{x^{4}}}$

Step 4.1.3

Since the exponent $x^{4}$ approaches $\infty$ , the quantity $e^{x^{4}}$ approaches $\infty$ .

$\frac{5}{4} \cdot \frac{\infty}{\infty}$

Step 4.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{5}{4} \cdot \frac{\infty}{\infty}$

Step 4.2

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

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

$\frac{5}{4} lim_{x \to \infty} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [e^{x^{4}}]}$

Step 4.3.2

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

$\frac{5}{4} lim_{x \to \infty} \frac{1}{\frac{d}{d x} [e^{x^{4}}]}$

Step 4.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{4}$ .

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Step 4.3.3.1

To apply the Chain Rule, set $u$ as $x^{4}$ .

$\frac{5}{4} lim_{x \to \infty} \frac{1}{\frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{4}]}$

Step 4.3.3.2

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

$\frac{5}{4} lim_{x \to \infty} \frac{1}{e^{u} \frac{d}{d x} [x^{4}]}$

Step 4.3.3.3

Replace all occurrences of $u$ with $x^{4}$ .

$\frac{5}{4} lim_{x \to \infty} \frac{1}{e^{x^{4}} \frac{d}{d x} [x^{4}]}$

Step 4.3.4

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

$\frac{5}{4} lim_{x \to \infty} \frac{1}{e^{x^{4}} \cdot 4 x^{3}}$

Step 4.3.5

Simplify.

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Step 4.3.5.1

Reorder the factors of $e^{x^{4}} (4 x^{3})$ .

$\frac{5}{4} lim_{x \to \infty} \frac{1}{4 e^{x^{4}} x^{3}}$

Step 4.3.5.2

Reorder factors in $4 e^{x^{4}} x^{3}$ .

$\frac{5}{4} lim_{x \to \infty} \frac{1}{4 x^{3} e^{x^{4}}}$

Step 5

Move the term $\frac{1}{4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{5}{4} \cdot \frac{1}{4} lim_{x \to \infty} \frac{1}{x^{3} e^{x^{4}}}$

Step 6

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{3} e^{x^{4}}}$ approaches $0$ .

$\frac{5}{4} \cdot \frac{1}{4} \cdot 0$

Step 7

Simplify the answer.

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Step 7.1

Multiply $\frac{5}{4} \cdot \frac{1}{4}$ .

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Step 7.1.1

Multiply $\frac{5}{4}$ by $\frac{1}{4}$ .

$\frac{5}{4 \cdot 4} \cdot 0$

Step 7.1.2

Multiply $4$ by $4$ .

$\frac{5}{16} \cdot 0$

Step 7.2

Multiply $\frac{5}{16}$ by $0$ .

$0$