Calculus Examples

$lim_{x \to \infty} \frac{x^{3}}{7 e^{\frac{x}{2}}}$

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.

$\frac{lim_{x \to \infty} x^{3}}{lim_{x \to \infty} 7 e^{\frac{x}{2}}}$

Step 1.2

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

$\frac{\infty}{lim_{x \to \infty} 7 e^{\frac{x}{2}}}$

Step 1.3

Since the function $e^{\frac{1}{2} x}$ approaches $\infty$ , the positive constant $7$ times the function also approaches $\infty$ .

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Step 1.3.1

Consider the limit with the constant multiple $7$ removed.

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

Step 1.3.2

Since the exponent $\frac{1}{2} x$ approaches $\infty$ , the quantity $e^{\frac{1}{2} x}$ approaches $\infty$ .

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

Step 1.3.3

Infinity divided by infinity is undefined.

Undefined

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

Step 1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 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.

$lim_{x \to \infty} \frac{x^{3}}{7 e^{\frac{x}{2}}} = lim_{x \to \infty} \frac{\frac{d}{d x} [x^{3}]}{\frac{d}{d x} [7 e^{\frac{x}{2}}]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [x^{3}]}{\frac{d}{d x} [7 e^{\frac{x}{2}}]}$

Step 3.2

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

$lim_{x \to \infty} \frac{3 x^{2}}{\frac{d}{d x} [7 e^{\frac{x}{2}}]}$

Step 3.3

Since $7$ is constant with respect to $x$ , the derivative of $7 e^{\frac{x}{2}}$ with respect to $x$ is $7 \frac{d}{d x} [e^{\frac{x}{2}}]$ .

$lim_{x \to \infty} \frac{3 x^{2}}{7 \frac{d}{d x} [e^{\frac{x}{2}}]}$

Step 3.4

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 $g (x) = \frac{x}{2}$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $\frac{x}{2}$ .

$lim_{x \to \infty} \frac{3 x^{2}}{7 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [\frac{x}{2}])}$

Step 3.4.2

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

$lim_{x \to \infty} \frac{3 x^{2}}{7 (e^{u} \frac{d}{d x} [\frac{x}{2}])}$

Step 3.4.3

Replace all occurrences of $u$ with $\frac{x}{2}$ .

$lim_{x \to \infty} \frac{3 x^{2}}{7 (e^{\frac{x}{2}} \frac{d}{d x} [\frac{x}{2}])}$

Step 3.5

Remove parentheses.

$lim_{x \to \infty} \frac{3 x^{2}}{7 e^{\frac{x}{2}} \frac{d}{d x} [\frac{x}{2}]}$

Step 3.6

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$lim_{x \to \infty} \frac{3 x^{2}}{7 e^{\frac{x}{2}} (\frac{1}{2} \frac{d}{d x} [x])}$

Step 3.7

Combine $\frac{1}{2}$ and $7$ .

$lim_{x \to \infty} \frac{3 x^{2}}{\frac{7}{2} e^{\frac{x}{2}} \frac{d}{d x} [x]}$

Step 3.8

Combine $\frac{7}{2}$ and $e^{\frac{x}{2}}$ .

$lim_{x \to \infty} \frac{3 x^{2}}{\frac{7 e^{\frac{x}{2}}}{2} \frac{d}{d x} [x]}$

Step 3.9

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

$lim_{x \to \infty} \frac{3 x^{2}}{\frac{7 e^{\frac{x}{2}}}{2} \cdot 1}$

Step 3.10

Multiply $\frac{7 e^{\frac{x}{2}}}{2}$ by $1$ .

$lim_{x \to \infty} \frac{3 x^{2}}{\frac{7 e^{\frac{x}{2}}}{2}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to \infty} 3 x^{2} \frac{2}{7 e^{\frac{x}{2}}}$

Step 5

Combine factors.

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Step 5.1

Combine $3$ and $\frac{2}{7 e^{\frac{x}{2}}}$ .

$lim_{x \to \infty} x^{2} \frac{3 \cdot 2}{7 e^{\frac{x}{2}}}$

Step 5.2

Multiply $3$ by $2$ .

$lim_{x \to \infty} x^{2} \frac{6}{7 e^{\frac{x}{2}}}$

Step 5.3

Combine $x^{2}$ and $\frac{6}{7 e^{\frac{x}{2}}}$ .

$lim_{x \to \infty} \frac{x^{2} \cdot 6}{7 e^{\frac{x}{2}}}$

Step 6

Move the term $\frac{6}{7}$ outside of the limit because it is constant with respect to $x$ .

$\frac{6}{7} lim_{x \to \infty} \frac{x^{2}}{e^{\frac{x}{2}}}$

Step 7

Apply L'Hospital's rule.

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Step 7.1

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

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Step 7.1.1

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

$\frac{6}{7} \cdot \frac{lim_{x \to \infty} x^{2}}{lim_{x \to \infty} e^{\frac{x}{2}}}$

Step 7.1.2

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

$\frac{6}{7} \cdot \frac{\infty}{lim_{x \to \infty} e^{\frac{x}{2}}}$

Step 7.1.3

Since the exponent $\frac{x}{2}$ approaches $\infty$ , the quantity $e^{\frac{x}{2}}$ approaches $\infty$ .

$\frac{6}{7} \cdot \frac{\infty}{\infty}$

Step 7.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{6}{7} \cdot \frac{\infty}{\infty}$

Step 7.2

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

Step 7.3

Find the derivative of the numerator and denominator.

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Step 7.3.1

Differentiate the numerator and denominator.

$\frac{6}{7} lim_{x \to \infty} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [e^{\frac{x}{2}}]}$

Step 7.3.2

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

$\frac{6}{7} lim_{x \to \infty} \frac{2 x}{\frac{d}{d x} [e^{\frac{x}{2}}]}$

Step 7.3.3

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 $g (x) = \frac{x}{2}$ .

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Step 7.3.3.1

To apply the Chain Rule, set $u$ as $\frac{x}{2}$ .

$\frac{6}{7} lim_{x \to \infty} \frac{2 x}{\frac{d}{d u} [e^{u}] \frac{d}{d x} [\frac{x}{2}]}$

Step 7.3.3.2

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

$\frac{6}{7} lim_{x \to \infty} \frac{2 x}{e^{u} \frac{d}{d x} [\frac{x}{2}]}$

Step 7.3.3.3

Replace all occurrences of $u$ with $\frac{x}{2}$ .

$\frac{6}{7} lim_{x \to \infty} \frac{2 x}{e^{\frac{x}{2}} \frac{d}{d x} [\frac{x}{2}]}$

Step 7.3.4

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$\frac{6}{7} lim_{x \to \infty} \frac{2 x}{e^{\frac{x}{2}} \frac{1}{2} \frac{d}{d x} [x]}$

Step 7.3.5

Combine $\frac{1}{2}$ and $e^{\frac{x}{2}}$ .

$\frac{6}{7} lim_{x \to \infty} \frac{2 x}{\frac{e^{\frac{x}{2}}}{2} \frac{d}{d x} [x]}$

Step 7.3.6

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

$\frac{6}{7} lim_{x \to \infty} \frac{2 x}{\frac{e^{\frac{x}{2}}}{2} \cdot 1}$

Step 7.3.7

Multiply $\frac{e^{\frac{x}{2}}}{2}$ by $1$ .

$\frac{6}{7} lim_{x \to \infty} \frac{2 x}{\frac{e^{\frac{x}{2}}}{2}}$

Step 7.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{6}{7} lim_{x \to \infty} 2 x \frac{2}{e^{\frac{x}{2}}}$

Step 7.5

Combine factors.

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Step 7.5.1

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

$\frac{6}{7} lim_{x \to \infty} x \frac{2 \cdot 2}{e^{\frac{x}{2}}}$

Step 7.5.2

Multiply $2$ by $2$ .

$\frac{6}{7} lim_{x \to \infty} x \frac{4}{e^{\frac{x}{2}}}$

Step 7.5.3

Combine $x$ and $\frac{4}{e^{\frac{x}{2}}}$ .

$\frac{6}{7} lim_{x \to \infty} \frac{x \cdot 4}{e^{\frac{x}{2}}}$

Step 8

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{6}{7} \cdot 4 lim_{x \to \infty} \frac{x}{e^{\frac{x}{2}}}$

Step 9

Apply L'Hospital's rule.

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Step 9.1

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

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Step 9.1.1

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

$\frac{6}{7} \cdot 4 \frac{lim_{x \to \infty} x}{lim_{x \to \infty} e^{\frac{x}{2}}}$

Step 9.1.2

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

$\frac{6}{7} \cdot 4 \frac{\infty}{lim_{x \to \infty} e^{\frac{x}{2}}}$

Step 9.1.3

Since the exponent $\frac{x}{2}$ approaches $\infty$ , the quantity $e^{\frac{x}{2}}$ approaches $\infty$ .

$\frac{6}{7} \cdot 4 \frac{\infty}{\infty}$

Step 9.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{6}{7} \cdot 4 \frac{\infty}{\infty}$

Step 9.2

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

Step 9.3

Find the derivative of the numerator and denominator.

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Step 9.3.1

Differentiate the numerator and denominator.

$\frac{6}{7} \cdot 4 lim_{x \to \infty} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [e^{\frac{x}{2}}]}$

Step 9.3.2

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

$\frac{6}{7} \cdot 4 lim_{x \to \infty} \frac{1}{\frac{d}{d x} [e^{\frac{x}{2}}]}$

Step 9.3.3

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 $g (x) = \frac{x}{2}$ .

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Step 9.3.3.1

To apply the Chain Rule, set $u$ as $\frac{x}{2}$ .

$\frac{6}{7} \cdot 4 lim_{x \to \infty} \frac{1}{\frac{d}{d u} [e^{u}] \frac{d}{d x} [\frac{x}{2}]}$

Step 9.3.3.2

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

$\frac{6}{7} \cdot 4 lim_{x \to \infty} \frac{1}{e^{u} \frac{d}{d x} [\frac{x}{2}]}$

Step 9.3.3.3

Replace all occurrences of $u$ with $\frac{x}{2}$ .

$\frac{6}{7} \cdot 4 lim_{x \to \infty} \frac{1}{e^{\frac{x}{2}} \frac{d}{d x} [\frac{x}{2}]}$

Step 9.3.4

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$\frac{6}{7} \cdot 4 lim_{x \to \infty} \frac{1}{e^{\frac{x}{2}} \frac{1}{2} \frac{d}{d x} [x]}$

Step 9.3.5

Combine $\frac{1}{2}$ and $e^{\frac{x}{2}}$ .

$\frac{6}{7} \cdot 4 lim_{x \to \infty} \frac{1}{\frac{e^{\frac{x}{2}}}{2} \frac{d}{d x} [x]}$

Step 9.3.6

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

$\frac{6}{7} \cdot 4 lim_{x \to \infty} \frac{1}{\frac{e^{\frac{x}{2}}}{2} \cdot 1}$

Step 9.3.7

Multiply $\frac{e^{\frac{x}{2}}}{2}$ by $1$ .

$\frac{6}{7} \cdot 4 lim_{x \to \infty} \frac{1}{\frac{e^{\frac{x}{2}}}{2}}$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{6}{7} \cdot 4 lim_{x \to \infty} 1 \frac{2}{e^{\frac{x}{2}}}$

Step 9.5

Multiply $\frac{2}{e^{\frac{x}{2}}}$ by $1$ .

$\frac{6}{7} \cdot 4 lim_{x \to \infty} \frac{2}{e^{\frac{x}{2}}}$

Step 10

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

$\frac{6}{7} \cdot 4 \cdot 2 lim_{x \to \infty} \frac{1}{e^{\frac{x}{2}}}$

Step 11

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{e^{\frac{x}{2}}}$ approaches $0$ .

$\frac{6}{7} \cdot 4 \cdot 2 \cdot 0$

Step 12

Simplify the answer.

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Step 12.1

Multiply $\frac{6}{7} \cdot 4$ .

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Step 12.1.1

Combine $\frac{6}{7}$ and $4$ .

$\frac{6 \cdot 4}{7} \cdot 2 \cdot 0$

Step 12.1.2

Multiply $6$ by $4$ .

$\frac{24}{7} \cdot 2 \cdot 0$

Step 12.2

Multiply $\frac{24}{7} \cdot 2$ .

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Step 12.2.1

Combine $\frac{24}{7}$ and $2$ .

$\frac{24 \cdot 2}{7} \cdot 0$

Step 12.2.2

Multiply $24$ by $2$ .

$\frac{48}{7} \cdot 0$

Step 12.3

Multiply $\frac{48}{7}$ by $0$ .

$0$