Calculus Examples

$lim_{x \to \infty} \frac{\sqrt[x]{3}}{9}$

Step 1

Evaluate the limit.

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Step 1.1

Move the term $\frac{1}{9}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{9} lim_{x \to \infty} \sqrt[x]{3}$

Step 1.2

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

$\frac{1}{9} lim_{x \to \infty} 3^{\frac{1}{x}}$

Step 1.3

Move the limit into the exponent.

$\frac{1}{9} \cdot 3^{lim_{x \to \infty} \frac{1}{x}}$

Step 2

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

$\frac{1}{9} \cdot 3^{0}$

Step 3

Simplify the answer.

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Step 3.1

Anything raised to $0$ is $1$ .

$\frac{1}{9} \cdot 1$

Step 3.2

Multiply $\frac{1}{9}$ by $1$ .

$\frac{1}{9}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{9}$

Decimal Form:

$0.\overline{1}$