Calculus Examples

$lim_{x \to 8} (\sqrt{x^{2} + 3 x}) - \sqrt{x^{2} + x}$

Step 1

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

$lim_{x \to 8} \sqrt{x^{2} + 3 x} - lim_{x \to 8} \sqrt{x^{2} + x}$

Step 2

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} x^{2} + 3 x} - lim_{x \to 8} \sqrt{x^{2} + x}$

Step 3

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

$\sqrt{lim_{x \to 8} x^{2} + lim_{x \to 8} 3 x} - lim_{x \to 8} \sqrt{x^{2} + x}$

Step 4

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

$\sqrt{{(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 3 x} - lim_{x \to 8} \sqrt{x^{2} + x}$

Step 5

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

$\sqrt{{(lim_{x \to 8} x)}^{2} + 3 lim_{x \to 8} x} - lim_{x \to 8} \sqrt{x^{2} + x}$

Step 6

Move the limit under the radical sign.

$\sqrt{{(lim_{x \to 8} x)}^{2} + 3 lim_{x \to 8} x} - \sqrt{lim_{x \to 8} x^{2} + x}$

Step 7

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

$\sqrt{{(lim_{x \to 8} x)}^{2} + 3 lim_{x \to 8} x} - \sqrt{lim_{x \to 8} x^{2} + lim_{x \to 8} x}$

Step 8

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

$\sqrt{{(lim_{x \to 8} x)}^{2} + 3 lim_{x \to 8} x} - \sqrt{{(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x}$

Step 9

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

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Step 9.1

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

$\sqrt{8^{2} + 3 lim_{x \to 8} x} - \sqrt{{(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x}$

Step 9.2

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

$\sqrt{8^{2} + 3 \cdot 8} - \sqrt{{(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x}$

Step 9.3

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

$\sqrt{8^{2} + 3 \cdot 8} - \sqrt{8^{2} + lim_{x \to 8} x}$

Step 9.4

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

$\sqrt{8^{2} + 3 \cdot 8} - \sqrt{8^{2} + 8}$

Step 10

Simplify each term.

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Step 10.1

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

$\sqrt{64 + 3 \cdot 8} - \sqrt{8^{2} + 8}$

Step 10.2

Multiply $3$ by $8$ .

$\sqrt{64 + 24} - \sqrt{8^{2} + 8}$

Step 10.3

Add $64$ and $24$ .

$\sqrt{88} - \sqrt{8^{2} + 8}$

Step 10.4

Rewrite $88$ as $2^{2} \cdot 22$ .

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Step 10.4.1

Factor $4$ out of $88$ .

$\sqrt{4 (22)} - \sqrt{8^{2} + 8}$

Step 10.4.2

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

$\sqrt{2^{2} \cdot 22} - \sqrt{8^{2} + 8}$

Step 10.5

Pull terms out from under the radical.

$2 \sqrt{22} - \sqrt{8^{2} + 8}$

Step 10.6

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

$2 \sqrt{22} - \sqrt{64 + 8}$

Step 10.7

Add $64$ and $8$ .

$2 \sqrt{22} - \sqrt{72}$

Step 10.8

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

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Step 10.8.1

Factor $36$ out of $72$ .

$2 \sqrt{22} - \sqrt{36 (2)}$

Step 10.8.2

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

$2 \sqrt{22} - \sqrt{6^{2} \cdot 2}$

Step 10.9

Pull terms out from under the radical.

$2 \sqrt{22} - (6 \sqrt{2})$

Step 10.10

Multiply $6$ by $- 1$ .

$2 \sqrt{22} - 6 \sqrt{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{22} - 6 \sqrt{2}$

Decimal Form:

$0.89555014 \dots$