Calculus Examples

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

Step 1

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

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

Step 2

Move the limit under the radical sign.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

$\sqrt{8^{2} + 8} - 8$

Step 6

Simplify each term.

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Step 6.1

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

$\sqrt{64 + 8} - 8$

Step 6.2

Add $64$ and $8$ .

$\sqrt{72} - 8$

Step 6.3

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

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Step 6.3.1

Factor $36$ out of $72$ .

$\sqrt{36 (2)} - 8$

Step 6.3.2

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

$\sqrt{6^{2} \cdot 2} - 8$

Step 6.4

Pull terms out from under the radical.

$6 \sqrt{2} - 8$

Step 7

The result can be shown in multiple forms.

Exact Form:

$6 \sqrt{2} - 8$

Decimal Form:

$0.48528137 \dots$