Calculus Examples

$\frac{lim_{x \to 0} \sqrt{x + 2} - \sqrt{2}}{x}$

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 $0$ .

$\frac{lim_{x \to 0} \sqrt{x + 2} - lim_{x \to 0} \sqrt{2}}{x}$

Step 1.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 0} x + 2} - lim_{x \to 0} \sqrt{2}}{x}$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\sqrt{lim_{x \to 0} x + lim_{x \to 0} 2} - lim_{x \to 0} \sqrt{2}}{x}$

Step 1.4

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$\frac{\sqrt{lim_{x \to 0} x + 2} - lim_{x \to 0} \sqrt{2}}{x}$

Step 1.5

Evaluate the limit of $\sqrt{2}$ which is constant as $x$ approaches $0$ .

$\frac{\sqrt{lim_{x \to 0} x + 2} - \sqrt{2}}{x}$

Step 2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $0$ and $2$ .

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

Step 3.1.2

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

$\frac{0}{x}$

Step 3.2

Divide $0$ by $x$ .

$0$