Calculus Examples

$lim_{n \to 8} \frac{1 - {(- 1)}^{n}}{\sqrt{n}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $n$ approaches $8$ .

$\frac{lim_{n \to 8} 1 - {(- 1)}^{n}}{lim_{n \to 8} \sqrt{n}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $n$ approaches $8$ .

$\frac{lim_{n \to 8} 1 - lim_{n \to 8} {(- 1)}^{n}}{lim_{n \to 8} \sqrt{n}}$

Step 3

Evaluate the limit of $1$ which is constant as $n$ approaches $8$ .

$\frac{1 - lim_{n \to 8} {(- 1)}^{n}}{lim_{n \to 8} \sqrt{n}}$

Step 4

Move the limit into the exponent.

$\frac{1 - {(- 1)}^{lim_{n \to 8} n}}{lim_{n \to 8} \sqrt{n}}$

Step 5

Move the limit under the radical sign.

$\frac{1 - {(- 1)}^{lim_{n \to 8} n}}{\sqrt{lim_{n \to 8} n}}$

Step 6

Evaluate the limits by plugging in $8$ for all occurrences of $n$ .

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Step 6.1

Evaluate the limit of $n$ by plugging in $8$ for $n$ .

$\frac{1 - {(- 1)}^{8}}{\sqrt{lim_{n \to 8} n}}$

Step 6.2

Evaluate the limit of $n$ by plugging in $8$ for $n$ .

$\frac{1 - {(- 1)}^{8}}{\sqrt{8}}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Multiply $- 1$ by ${(- 1)}^{8}$ by adding the exponents.

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Step 7.1.1.1

Multiply $- 1$ by ${(- 1)}^{8}$ .

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Step 7.1.1.1.1

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

$\frac{1 + {(- 1)}^{1} {(- 1)}^{8}}{\sqrt{8}}$

Step 7.1.1.1.2

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

$\frac{1 + {(- 1)}^{1 + 8}}{\sqrt{8}}$

Step 7.1.1.2

Add $1$ and $8$ .

$\frac{1 + {(- 1)}^{9}}{\sqrt{8}}$

Step 7.1.2

Raise $- 1$ to the power of $9$ .

$\frac{1 - 1}{\sqrt{8}}$

Step 7.1.3

Subtract $1$ from $1$ .

$\frac{0}{\sqrt{8}}$

Step 7.2

Simplify the denominator.

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Step 7.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 7.2.1.1

Factor $4$ out of $8$ .

$\frac{0}{\sqrt{4 (2)}}$

Step 7.2.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 7.2.2

Pull terms out from under the radical.

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

Step 7.3

Cancel the common factor of $0$ and $2$ .

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Step 7.3.1

Factor $2$ out of $0$ .

$\frac{2 (0)}{2 \sqrt{2}}$

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

Factor $2$ out of $2 \sqrt{2}$ .

$\frac{2 (0)}{2 (\sqrt{2})}$

Step 7.3.2.2

Cancel the common factor.

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

Step 7.3.2.3

Rewrite the expression.

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

Step 7.4

Divide $0$ by $\sqrt{2}$ .

$0$