Calculus Examples

$lim_{x \to \infty} \frac{(\ln (\sqrt{x}))}{x^{2}}$

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.

$\frac{lim_{x \to \infty} \ln (\sqrt{x})}{lim_{x \to \infty} x^{2}}$

Step 1.2

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

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

Step 1.3

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

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

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.

$lim_{x \to \infty} \frac{(\ln (\sqrt{x}))}{x^{2}} = lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (\sqrt{x})]}{\frac{d}{d x} [x^{2}]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x^{\frac{1}{2}}$ .

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Step 3.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{2}}$ .

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

Step 3.3.2

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

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

Step 3.3.3

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Combine the numerators over the common denominator.

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

Step 3.8

Simplify the numerator.

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Step 3.8.1

Multiply $- 1$ by $2$ .

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

Step 3.8.2

Subtract $2$ from $1$ .

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

Step 3.9

Move the negative in front of the fraction.

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

Step 3.10

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

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

Step 3.11

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

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

Step 3.12

Move $2$ to the left of $x^{\frac{1}{2}}$ .

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

Step 3.13

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

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

Step 3.14

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

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

Step 3.15

Simplify the denominator.

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Step 3.15.1

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

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Step 3.15.1.1

Move $x^{\frac{1}{2}}$ .

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

Step 3.15.1.2

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

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

Step 3.15.1.3

Combine the numerators over the common denominator.

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

Step 3.15.1.4

Add $1$ and $1$ .

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

Step 3.15.1.5

Divide $2$ by $2$ .

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

Step 3.15.2

Simplify $2 x^{1}$ .

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

Step 3.16

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine factors.

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Step 5.1

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

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

Step 5.2

Multiply $2$ by $2$ .

$lim_{x \to \infty} \frac{1}{4 x \cdot x}$

Step 5.3

Raise $x$ to the power of $1$ .

$lim_{x \to \infty} \frac{1}{4 (x^{1} x)}$

Step 5.4

Raise $x$ to the power of $1$ .

$lim_{x \to \infty} \frac{1}{4 (x^{1} x^{1})}$

Step 5.5

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

$lim_{x \to \infty} \frac{1}{4 x^{1 + 1}}$

Step 5.6

Add $1$ and $1$ .

$lim_{x \to \infty} \frac{1}{4 x^{2}}$

Step 6

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

$\frac{1}{4} lim_{x \to \infty} \frac{1}{x^{2}}$

Step 7

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

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

Step 8

Multiply $\frac{1}{4}$ by $0$ .

$0$