Calculus Examples

$\frac{lim_{x \to 2} \sqrt{x} - \sqrt{2}}{\sqrt{x^{2} - 4}}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} \sqrt{x} - lim_{x \to 2} \sqrt{2}}{\sqrt{x^{2} - 4}}$

Step 1.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 2} x} - lim_{x \to 2} \sqrt{2}}{\sqrt{x^{2} - 4}}$

Step 1.3

Evaluate the limit of $\sqrt{2}$ which is constant as $x$ approaches $2$ .

$\frac{\sqrt{lim_{x \to 2} x} - \sqrt{2}}{\sqrt{x^{2} - 4}}$

Step 2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{\sqrt{2} - \sqrt{2}}{\sqrt{x^{2} - 4}}$

Step 3

Simplify the answer.

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Step 3.1

Subtract $\sqrt{2}$ from $\sqrt{2}$ .

$\frac{0}{\sqrt{x^{2} - 4}}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

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

$\frac{0}{\sqrt{x^{2} - 2^{2}}}$

Step 3.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{0}{\sqrt{(x + 2) (x - 2)}}$

Step 3.3

Divide $0$ by $\sqrt{(x + 2) (x - 2)}$ .

$0$