Calculus Examples

$\frac{lim_{n \to 8} \sqrt{n^{2} + n}}{n}$

Step 1

Move the limit under the radical sign.

$\frac{\sqrt{lim_{n \to 8} n^{2} + n}}{n}$

Step 2

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

$\frac{\sqrt{lim_{n \to 8} n^{2} + lim_{n \to 8} n}}{n}$

Step 3

Move the exponent $2$ from $n^{2}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt{{(lim_{n \to 8} n)}^{2} + lim_{n \to 8} n}}{n}$

Step 4

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

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Step 4.1

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

$\frac{\sqrt{8^{2} + lim_{n \to 8} n}}{n}$

Step 4.2

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

$\frac{\sqrt{8^{2} + 8}}{n}$

Step 5

Simplify the numerator.

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Step 5.1

Raise $8$ to the power of $2$ .

$\frac{\sqrt{64 + 8}}{n}$

Step 5.2

Add $64$ and $8$ .

$\frac{\sqrt{72}}{n}$

Step 5.3

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 5.3.1

Factor $36$ out of $72$ .

$\frac{\sqrt{36 (2)}}{n}$

Step 5.3.2

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{6^{2} \cdot 2}}{n}$

Step 5.4

Pull terms out from under the radical.

$\frac{6 \sqrt{2}}{n}$