Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

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.

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

Step 3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3.6

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

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

Step 3.7

Combine the numerators over the common denominator.

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

Step 3.8

Simplify the numerator.

Tap for more steps...

Step 3.8.1

Multiply $- 1$ by $3$ .

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

Step 3.8.2

Subtract $3$ from $1$ .

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

Step 3.9

Move the negative in front of the fraction.

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

Step 3.10

Simplify.

Tap for more steps...

Step 3.10.1

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

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

Step 3.10.2

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine factors.

Tap for more steps...

Step 5.1

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

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

Step 5.2

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

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

Step 6

Reduce.

Tap for more steps...

Step 6.1

Factor $x$ out of $3 x^{\frac{2}{3}}$ .

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

Step 6.2

Cancel the common factors.

Tap for more steps...

Step 6.2.1

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

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

Step 6.2.2

Factor $x$ out of $x^{1}$ .

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

Step 6.2.3

Cancel the common factor.

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

Step 6.2.4

Rewrite the expression.

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

Step 6.2.5

Divide $3 x^{- \frac{1}{3}}$ by $1$ .

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

Step 7

Evaluate the limit.

Tap for more steps...

Step 7.1

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

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

Step 7.2

Simplify the limit argument.

Tap for more steps...

Step 7.2.1

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

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

Step 7.2.2

Rewrite $x^{\frac{1}{3}}$ as $\sqrt[3]{x}$ .

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

Step 8

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{\sqrt[3]{x}}$ approaches $0$ .

$3 \cdot 0$

Step 9

Multiply $3$ by $0$ .

$0$