Precalculus Examples

$lim_{x \to 8} \sqrt{x + \sqrt{x}} - \sqrt{x}$

Step 1

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

$lim_{x \to 8} \sqrt{x + \sqrt{x}} - lim_{x \to 8} \sqrt{x}$

Step 2

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} x + \sqrt{x}} - lim_{x \to 8} \sqrt{x}$

Step 3

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

$\sqrt{lim_{x \to 8} x + lim_{x \to 8} \sqrt{x}} - lim_{x \to 8} \sqrt{x}$

Step 4

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} x + \sqrt{lim_{x \to 8} x}} - lim_{x \to 8} \sqrt{x}$

Step 5

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} x + \sqrt{lim_{x \to 8} x}} - \sqrt{lim_{x \to 8} x}$

Step 6

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

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Step 6.1

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

$\sqrt{8 + \sqrt{lim_{x \to 8} x}} - \sqrt{lim_{x \to 8} x}$

Step 6.2

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

$\sqrt{8 + \sqrt{8}} - \sqrt{lim_{x \to 8} x}$

Step 6.3

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

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

Step 7

Simplify each term.

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Step 7.1

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

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Step 7.1.1

Factor $4$ out of $8$ .

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

Step 7.1.2

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

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

Step 7.2

Pull terms out from under the radical.

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

Step 7.3

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

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Step 7.3.1

Factor $4$ out of $8$ .

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

Step 7.3.2

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

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

Step 7.4

Pull terms out from under the radical.

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

Step 7.5

Multiply $2$ by $- 1$ .

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.46223042 \dots$