Calculus Examples

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

Step 1

Rewrite $x^{3} e^{- \frac{x}{5}}$ as $\frac{x^{3}}{e^{\frac{x}{5}}}$ .

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

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.

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

Step 2.1.2

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

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

Step 2.1.3

Since the exponent $\frac{x}{5}$ approaches $\infty$ , the quantity $e^{\frac{x}{5}}$ 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.

$lim_{x \to \infty} \frac{x^{3}}{e^{\frac{x}{5}}} = lim_{x \to \infty} \frac{\frac{d}{d x} [x^{3}]}{\frac{d}{d x} [e^{\frac{x}{5}}]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [x^{3}]}{\frac{d}{d x} [e^{\frac{x}{5}}]}$

Step 2.3.2

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

$lim_{x \to \infty} \frac{3 x^{2}}{\frac{d}{d x} [e^{\frac{x}{5}}]}$

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 $g (x) = \frac{x}{5}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u$ as $\frac{x}{5}$ .

$lim_{x \to \infty} \frac{3 x^{2}}{\frac{d}{d u} [e^{u}] \frac{d}{d x} [\frac{x}{5}]}$

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 $a$ = $e$ .

$lim_{x \to \infty} \frac{3 x^{2}}{e^{u} \frac{d}{d x} [\frac{x}{5}]}$

Step 2.3.3.3

Replace all occurrences of $u$ with $\frac{x}{5}$ .

$lim_{x \to \infty} \frac{3 x^{2}}{e^{\frac{x}{5}} \frac{d}{d x} [\frac{x}{5}]}$

Step 2.3.4

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{x}{5}$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [x]$ .

$lim_{x \to \infty} \frac{3 x^{2}}{e^{\frac{x}{5}} (\frac{1}{5} \frac{d}{d x} [x])}$

Step 2.3.5

Combine $\frac{1}{5}$ and $e^{\frac{x}{5}}$ .

$lim_{x \to \infty} \frac{3 x^{2}}{\frac{e^{\frac{x}{5}}}{5} \frac{d}{d x} [x]}$

Step 2.3.6

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

$lim_{x \to \infty} \frac{3 x^{2}}{\frac{e^{\frac{x}{5}}}{5} \cdot 1}$

Step 2.3.7

Multiply $\frac{e^{\frac{x}{5}}}{5}$ by $1$ .

$lim_{x \to \infty} \frac{3 x^{2}}{\frac{e^{\frac{x}{5}}}{5}}$

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to \infty} 3 x^{2} \frac{5}{e^{\frac{x}{5}}}$

Step 2.5

Combine factors.

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Step 2.5.1

Combine $3$ and $\frac{5}{e^{\frac{x}{5}}}$ .

$lim_{x \to \infty} x^{2} \frac{3 \cdot 5}{e^{\frac{x}{5}}}$

Step 2.5.2

Multiply $3$ by $5$ .

$lim_{x \to \infty} x^{2} \frac{15}{e^{\frac{x}{5}}}$

Step 2.5.3

Combine $x^{2}$ and $\frac{15}{e^{\frac{x}{5}}}$ .

$lim_{x \to \infty} \frac{x^{2} \cdot 15}{e^{\frac{x}{5}}}$

Step 2.6

Move $15$ to the left of $x^{2}$ .

$lim_{x \to \infty} \frac{15 x^{2}}{e^{\frac{x}{5}}}$

Step 3

Move the term $15$ outside of the limit because it is constant with respect to $x$ .

$15 lim_{x \to \infty} \frac{x^{2}}{e^{\frac{x}{5}}}$

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.

$15 \frac{lim_{x \to \infty} x^{2}}{lim_{x \to \infty} e^{\frac{x}{5}}}$

Step 4.1.2

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

$15 \frac{\infty}{lim_{x \to \infty} e^{\frac{x}{5}}}$

Step 4.1.3

Since the exponent $\frac{x}{5}$ approaches $\infty$ , the quantity $e^{\frac{x}{5}}$ approaches $\infty$ .

$15 \frac{\infty}{\infty}$

Step 4.1.4

Infinity divided by infinity is undefined.

Undefined

$15 \frac{\infty}{\infty}$

Step 4.2

$lim_{x \to \infty} \frac{x^{2}}{e^{\frac{x}{5}}} = lim_{x \to \infty} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [e^{\frac{x}{5}}]}$

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

$15 lim_{x \to \infty} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [e^{\frac{x}{5}}]}$

Step 4.3.2

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

$15 lim_{x \to \infty} \frac{2 x}{\frac{d}{d x} [e^{\frac{x}{5}}]}$

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 $g (x) = \frac{x}{5}$ .

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Step 4.3.3.1

To apply the Chain Rule, set $u$ as $\frac{x}{5}$ .

$15 lim_{x \to \infty} \frac{2 x}{\frac{d}{d u} [e^{u}] \frac{d}{d x} [\frac{x}{5}]}$

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 $a$ = $e$ .

$15 lim_{x \to \infty} \frac{2 x}{e^{u} \frac{d}{d x} [\frac{x}{5}]}$

Step 4.3.3.3

Replace all occurrences of $u$ with $\frac{x}{5}$ .

$15 lim_{x \to \infty} \frac{2 x}{e^{\frac{x}{5}} \frac{d}{d x} [\frac{x}{5}]}$

Step 4.3.4

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{x}{5}$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [x]$ .

$15 lim_{x \to \infty} \frac{2 x}{e^{\frac{x}{5}} \frac{1}{5} \frac{d}{d x} [x]}$

Step 4.3.5

Combine $\frac{1}{5}$ and $e^{\frac{x}{5}}$ .

$15 lim_{x \to \infty} \frac{2 x}{\frac{e^{\frac{x}{5}}}{5} \frac{d}{d x} [x]}$

Step 4.3.6

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

$15 lim_{x \to \infty} \frac{2 x}{\frac{e^{\frac{x}{5}}}{5} \cdot 1}$

Step 4.3.7

Multiply $\frac{e^{\frac{x}{5}}}{5}$ by $1$ .

$15 lim_{x \to \infty} \frac{2 x}{\frac{e^{\frac{x}{5}}}{5}}$

Step 4.4

Multiply the numerator by the reciprocal of the denominator.

$15 lim_{x \to \infty} 2 x \frac{5}{e^{\frac{x}{5}}}$

Step 4.5

Combine factors.

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Step 4.5.1

Combine $2$ and $\frac{5}{e^{\frac{x}{5}}}$ .

$15 lim_{x \to \infty} x \frac{2 \cdot 5}{e^{\frac{x}{5}}}$

Step 4.5.2

Multiply $2$ by $5$ .

$15 lim_{x \to \infty} x \frac{10}{e^{\frac{x}{5}}}$

Step 4.5.3

Combine $x$ and $\frac{10}{e^{\frac{x}{5}}}$ .

$15 lim_{x \to \infty} \frac{x \cdot 10}{e^{\frac{x}{5}}}$

Step 5

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$15 \cdot 10 lim_{x \to \infty} \frac{x}{e^{\frac{x}{5}}}$

Step 6

Apply L'Hospital's rule.

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Step 6.1

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

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Step 6.1.1

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

$15 \cdot 10 \frac{lim_{x \to \infty} x}{lim_{x \to \infty} e^{\frac{x}{5}}}$

Step 6.1.2

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

$15 \cdot 10 \frac{\infty}{lim_{x \to \infty} e^{\frac{x}{5}}}$

Step 6.1.3

Since the exponent $\frac{x}{5}$ approaches $\infty$ , the quantity $e^{\frac{x}{5}}$ approaches $\infty$ .

$15 \cdot 10 \frac{\infty}{\infty}$

Step 6.1.4

Infinity divided by infinity is undefined.

Undefined

$15 \cdot 10 \frac{\infty}{\infty}$

Step 6.2

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

Step 6.3

Find the derivative of the numerator and denominator.

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Step 6.3.1

Differentiate the numerator and denominator.

$15 \cdot 10 lim_{x \to \infty} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [e^{\frac{x}{5}}]}$

Step 6.3.2

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

$15 \cdot 10 lim_{x \to \infty} \frac{1}{\frac{d}{d x} [e^{\frac{x}{5}}]}$

Step 6.3.3

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 $g (x) = \frac{x}{5}$ .

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Step 6.3.3.1

To apply the Chain Rule, set $u$ as $\frac{x}{5}$ .

$15 \cdot 10 lim_{x \to \infty} \frac{1}{\frac{d}{d u} [e^{u}] \frac{d}{d x} [\frac{x}{5}]}$

Step 6.3.3.2

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

$15 \cdot 10 lim_{x \to \infty} \frac{1}{e^{u} \frac{d}{d x} [\frac{x}{5}]}$

Step 6.3.3.3

Replace all occurrences of $u$ with $\frac{x}{5}$ .

$15 \cdot 10 lim_{x \to \infty} \frac{1}{e^{\frac{x}{5}} \frac{d}{d x} [\frac{x}{5}]}$

Step 6.3.4

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{x}{5}$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [x]$ .

$15 \cdot 10 lim_{x \to \infty} \frac{1}{e^{\frac{x}{5}} \frac{1}{5} \frac{d}{d x} [x]}$

Step 6.3.5

Combine $\frac{1}{5}$ and $e^{\frac{x}{5}}$ .

$15 \cdot 10 lim_{x \to \infty} \frac{1}{\frac{e^{\frac{x}{5}}}{5} \frac{d}{d x} [x]}$

Step 6.3.6

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

$15 \cdot 10 lim_{x \to \infty} \frac{1}{\frac{e^{\frac{x}{5}}}{5} \cdot 1}$

Step 6.3.7

Multiply $\frac{e^{\frac{x}{5}}}{5}$ by $1$ .

$15 \cdot 10 lim_{x \to \infty} \frac{1}{\frac{e^{\frac{x}{5}}}{5}}$

Step 6.4

Multiply the numerator by the reciprocal of the denominator.

$15 \cdot 10 lim_{x \to \infty} 1 \frac{5}{e^{\frac{x}{5}}}$

Step 6.5

Multiply $\frac{5}{e^{\frac{x}{5}}}$ by $1$ .

$15 \cdot 10 lim_{x \to \infty} \frac{5}{e^{\frac{x}{5}}}$

Step 7

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$15 \cdot 10 \cdot 5 lim_{x \to \infty} \frac{1}{e^{\frac{x}{5}}}$

Step 8

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{e^{\frac{x}{5}}}$ approaches $0$ .

$15 \cdot 10 \cdot 5 \cdot 0$

Step 9

Multiply $15 \cdot 10 \cdot 5 \cdot 0$ .

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Step 9.1

Multiply $15$ by $10$ .

$150 \cdot 5 \cdot 0$

Step 9.2

Multiply $150$ by $5$ .

$750 \cdot 0$

Step 9.3

Multiply $750$ by $0$ .

$0$