Calculus Examples

$lim_{x \to \infty} \frac{\ln (3 x)}{\sqrt{3 x}}$

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.

$\frac{lim_{x \to \infty} \ln (3 x)}{lim_{x \to \infty} \sqrt{3 x}}$

Step 1.1.2

As log approaches infinity, the value goes to $\infty$ .

$\frac{\infty}{lim_{x \to \infty} \sqrt{3 x}}$

Step 1.1.3

As $x$ approaches $\infty$ for radicals, the value goes to $\infty$ .

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

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.

$lim_{x \to \infty} \frac{\ln (3 x)}{\sqrt{3 x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (3 x)]}{\frac{d}{d x} [\sqrt{3 x}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (3 x)]}{\frac{d}{d x} [\sqrt{3 x}]}$

Step 1.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 3 x$ .

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Step 1.3.2.1

To apply the Chain Rule, set $u$ as $3 x$ .

$lim_{x \to \infty} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 x]}{\frac{d}{d x} [\sqrt{3 x}]}$

Step 1.3.2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$lim_{x \to \infty} \frac{\frac{1}{u} \frac{d}{d x} [3 x]}{\frac{d}{d x} [\sqrt{3 x}]}$

Step 1.3.2.3

Replace all occurrences of $u$ with $3 x$ .

$lim_{x \to \infty} \frac{\frac{1}{3 x} \frac{d}{d x} [3 x]}{\frac{d}{d x} [\sqrt{3 x}]}$

Step 1.3.3

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

$lim_{x \to \infty} \frac{\frac{1}{3 x} (3 \frac{d}{d x} [x])}{\frac{d}{d x} [\sqrt{3 x}]}$

Step 1.3.4

Combine $3$ and $\frac{1}{3 x}$ .

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

Step 1.3.5

Cancel the common factor of $3$ .

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Step 1.3.5.1

Cancel the common factor.

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

Step 1.3.5.2

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{1}{x} \frac{d}{d x} [x]}{\frac{d}{d x} [\sqrt{3 x}]}$

Step 1.3.6

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

$lim_{x \to \infty} \frac{\frac{1}{x} \cdot 1}{\frac{d}{d x} [\sqrt{3 x}]}$

Step 1.3.7

Multiply $\frac{1}{x}$ by $1$ .

$lim_{x \to \infty} \frac{\frac{1}{x}}{\frac{d}{d x} [\sqrt{3 x}]}$

Step 1.3.8

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

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

Step 1.3.9

Factor $3$ out of $3 x$ .

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

Step 1.3.10

Apply the product rule to $3 (x)$ .

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

Step 1.3.11

Since $3^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $3^{\frac{1}{2}} x^{\frac{1}{2}}$ with respect to $x$ is $3^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

Step 1.3.12

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

$lim_{x \to \infty} \frac{\frac{1}{x}}{3^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1})}$

Step 1.3.13

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to \infty} \frac{\frac{1}{x}}{3^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})}$

Step 1.3.14

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

$lim_{x \to \infty} \frac{\frac{1}{x}}{3^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})}$

Step 1.3.15

Combine the numerators over the common denominator.

$lim_{x \to \infty} \frac{\frac{1}{x}}{3^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})}$

Step 1.3.16

Simplify the numerator.

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Step 1.3.16.1

Multiply $- 1$ by $2$ .

$lim_{x \to \infty} \frac{\frac{1}{x}}{3^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 2}{2}})}$

Step 1.3.16.2

Subtract $2$ from $1$ .

$lim_{x \to \infty} \frac{\frac{1}{x}}{3^{\frac{1}{2}} (\frac{1}{2} x^{\frac{- 1}{2}})}$

Step 1.3.17

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{\frac{1}{x}}{3^{\frac{1}{2}} (\frac{1}{2} x^{- \frac{1}{2}})}$

Step 1.3.18

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

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

Step 1.3.19

Combine $3^{\frac{1}{2}}$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

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

Step 1.3.20

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5

Convert fractional exponents to radicals.

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Step 1.5.1

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

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

Step 1.5.2

Rewrite $3^{\frac{1}{2}}$ as $\sqrt{3}$ .

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

Step 1.6

Multiply $\frac{1}{x}$ by $\frac{2 \sqrt{x}}{\sqrt{3}}$ .

$lim_{x \to \infty} \frac{2 \sqrt{x}}{x \sqrt{3}}$

Step 2

Move the term $\frac{2}{\sqrt{3}}$ outside of the limit because it is constant with respect to $x$ .

$\frac{2}{\sqrt{3}} lim_{x \to \infty} \frac{\sqrt{x}}{x}$

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.

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

Step 3.1.2

As $x$ approaches $\infty$ for radicals, the value goes to $\infty$ .

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

Step 3.1.3

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

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

Step 3.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 3.2

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 3.3.3

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

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

Step 3.3.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.3.5

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

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

Step 3.3.6

Combine the numerators over the common denominator.

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

Step 3.3.7

Simplify the numerator.

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Step 3.3.7.1

Multiply $- 1$ by $2$ .

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

Step 3.3.7.2

Subtract $2$ from $1$ .

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

Step 3.3.8

Move the negative in front of the fraction.

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

Step 3.3.9

Simplify.

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Step 3.3.9.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.3.9.2

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

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

Step 3.3.10

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

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

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

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

Step 3.6

Multiply $\frac{1}{2 \sqrt{x}}$ by $1$ .

$\frac{2}{\sqrt{3}} lim_{x \to \infty} \frac{1}{2 \sqrt{x}}$

Step 4

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

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

Step 5

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

$\frac{2}{\sqrt{3}} \cdot \frac{1}{2} \cdot 0$

Step 6

Simplify the answer.

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Step 6.1

Cancel the common factor of $2$ .

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Step 6.1.1

Cancel the common factor.

$\frac{2}{\sqrt{3}} \cdot \frac{1}{2} \cdot 0$

Step 6.1.2

Rewrite the expression.

$\frac{1}{\sqrt{3}} \cdot 0$

Step 6.2

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \cdot 0$

Step 6.3

Combine and simplify the denominator.

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Step 6.3.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} \cdot 0$

Step 6.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \cdot 0$

Step 6.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \cdot 0$

Step 6.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} \cdot 0$

Step 6.3.5

Add $1$ and $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{2}} \cdot 0$

Step 6.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \cdot 0$

Step 6.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \cdot 0$

Step 6.3.6.3

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

$\frac{\sqrt{3}}{3^{\frac{2}{2}}} \cdot 0$

Step 6.3.6.4

Cancel the common factor of $2$ .

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Step 6.3.6.4.1

Cancel the common factor.

$\frac{\sqrt{3}}{3^{\frac{2}{2}}} \cdot 0$

Step 6.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{3}}{3^{1}} \cdot 0$

Step 6.3.6.5

Evaluate the exponent.

$\frac{\sqrt{3}}{3} \cdot 0$

Step 6.4

Multiply $\frac{\sqrt{3}}{3}$ by $0$ .

$0$