Calculus Examples

$lim_{x \to \infty} \frac{\ln (5 x)}{\sqrt{5 x}}$

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.

$\frac{lim_{x \to \infty} \ln (5 x)}{lim_{x \to \infty} \sqrt{5 x}}$

Step 1.2

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

$\frac{\infty}{lim_{x \to \infty} \sqrt{5 x}}$

Step 1.3

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

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

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.

$lim_{x \to \infty} \frac{\ln (5 x)}{\sqrt{5 x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (5 x)]}{\frac{d}{d x} [\sqrt{5 x}]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (5 x)]}{\frac{d}{d x} [\sqrt{5 x}]}$

Step 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 $g (x) = 5 x$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $5 x$ .

$lim_{x \to \infty} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [5 x]}{\frac{d}{d x} [\sqrt{5 x}]}$

Step 3.2.2

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

$lim_{x \to \infty} \frac{\frac{1}{u} \frac{d}{d x} [5 x]}{\frac{d}{d x} [\sqrt{5 x}]}$

Step 3.2.3

Replace all occurrences of $u$ with $5 x$ .

$lim_{x \to \infty} \frac{\frac{1}{5 x} \frac{d}{d x} [5 x]}{\frac{d}{d x} [\sqrt{5 x}]}$

Step 3.3

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

$lim_{x \to \infty} \frac{\frac{1}{5 x} (5 \frac{d}{d x} [x])}{\frac{d}{d x} [\sqrt{5 x}]}$

Step 3.4

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

$lim_{x \to \infty} \frac{\frac{5}{5 x} \frac{d}{d x} [x]}{\frac{d}{d x} [\sqrt{5 x}]}$

Step 3.5

Cancel the common factor of $5$ .

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Step 3.5.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{5}{5 x} \frac{d}{d x} [x]}{\frac{d}{d x} [\sqrt{5 x}]}$

Step 3.5.2

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{1}{x} \frac{d}{d x} [x]}{\frac{d}{d x} [\sqrt{5 x}]}$

Step 3.6

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

$lim_{x \to \infty} \frac{\frac{1}{x} \cdot 1}{\frac{d}{d x} [\sqrt{5 x}]}$

Step 3.7

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

$lim_{x \to \infty} \frac{\frac{1}{x}}{\frac{d}{d x} [\sqrt{5 x}]}$

Step 3.8

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x}$ as ${(5 x)}^{\frac{1}{2}}$ .

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

Step 3.9

Factor $5$ out of $5 x$ .

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

Step 3.10

Apply the product rule to $5 (x)$ .

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

Step 3.11

Since $5^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $5^{\frac{1}{2}} x^{\frac{1}{2}}$ with respect to $x$ is $5^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

Step 3.12

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

$lim_{x \to \infty} \frac{\frac{1}{x}}{5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1})}$

Step 3.13

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

$lim_{x \to \infty} \frac{\frac{1}{x}}{5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})}$

Step 3.14

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

$lim_{x \to \infty} \frac{\frac{1}{x}}{5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})}$

Step 3.15

Combine the numerators over the common denominator.

$lim_{x \to \infty} \frac{\frac{1}{x}}{5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})}$

Step 3.16

Simplify the numerator.

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Step 3.16.1

Multiply $- 1$ by $2$ .

$lim_{x \to \infty} \frac{\frac{1}{x}}{5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 2}{2}})}$

Step 3.16.2

Subtract $2$ from $1$ .

$lim_{x \to \infty} \frac{\frac{1}{x}}{5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{- 1}{2}})}$

Step 3.17

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{\frac{1}{x}}{5^{\frac{1}{2}} (\frac{1}{2} x^{- \frac{1}{2}})}$

Step 3.18

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

$lim_{x \to \infty} \frac{\frac{1}{x}}{5^{\frac{1}{2}} \frac{x^{- \frac{1}{2}}}{2}}$

Step 3.19

Combine $5^{\frac{1}{2}}$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$lim_{x \to \infty} \frac{\frac{1}{x}}{\frac{5^{\frac{1}{2}} x^{- \frac{1}{2}}}{2}}$

Step 3.20

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Convert fractional exponents to radicals.

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Step 5.1

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

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

Step 5.2

Rewrite $5^{\frac{1}{2}}$ as $\sqrt{5}$ .

$lim_{x \to \infty} \frac{1}{x} \cdot \frac{2 \sqrt{x}}{\sqrt{5}}$

Step 6

Multiply $\frac{1}{x}$ by $\frac{2 \sqrt{x}}{\sqrt{5}}$ .

$lim_{x \to \infty} \frac{2 \sqrt{x}}{x \sqrt{5}}$

Step 7

Reduce.

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Step 7.1

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

$lim_{x \to \infty} \frac{2 x^{\frac{1}{2}}}{x \sqrt{5}}$

Step 7.2

Factor $x$ out of $2 x^{\frac{1}{2}}$ .

$lim_{x \to \infty} \frac{x (2 x^{- \frac{1}{2}})}{x \sqrt{5}}$

Step 7.3

Cancel the common factors.

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Step 7.3.1

Factor $x$ out of $x \sqrt{5}$ .

$lim_{x \to \infty} \frac{x (2 x^{- \frac{1}{2}})}{x (\sqrt{5})}$

Step 7.3.2

Cancel the common factor.

$lim_{x \to \infty} \frac{x (2 x^{- \frac{1}{2}})}{x \sqrt{5}}$

Step 7.3.3

Rewrite the expression.

$lim_{x \to \infty} \frac{2 x^{- \frac{1}{2}}}{\sqrt{5}}$

Step 7.4

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

$lim_{x \to \infty} \frac{2}{\sqrt{5} x^{\frac{1}{2}}}$

Step 8

Move the term $\frac{2}{\sqrt{5}}$ outside of the limit because it is constant with respect to $x$ .

$\frac{2}{\sqrt{5}} lim_{x \to \infty} \frac{1}{x^{\frac{1}{2}}}$

Step 9

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

$\frac{2}{\sqrt{5}} \cdot 0$

Step 10

Simplify the answer.

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Step 10.1

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} \cdot 0$

Step 10.2

Combine and simplify the denominator.

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Step 10.2.1

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}} \cdot 0$

Step 10.2.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{2 \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}} \cdot 0$

Step 10.2.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{2 \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}} \cdot 0$

Step 10.2.4

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

$\frac{2 \sqrt{5}}{{\sqrt{5}}^{1 + 1}} \cdot 0$

Step 10.2.5

Add $1$ and $1$ .

$\frac{2 \sqrt{5}}{{\sqrt{5}}^{2}} \cdot 0$

Step 10.2.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 10.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{2 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}} \cdot 0$

Step 10.2.6.2

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

$\frac{2 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}} \cdot 0$

Step 10.2.6.3

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

$\frac{2 \sqrt{5}}{5^{\frac{2}{2}}} \cdot 0$

Step 10.2.6.4

Cancel the common factor of $2$ .

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Step 10.2.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{5}}{5^{\frac{2}{2}}} \cdot 0$

Step 10.2.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{5}}{5^{1}} \cdot 0$

Step 10.2.6.5

Evaluate the exponent.

$\frac{2 \sqrt{5}}{5} \cdot 0$

Step 10.3

Multiply $\frac{2 \sqrt{5}}{5}$ by $0$ .

$0$