Calculus Examples

$lim_{x \to 0} \sqrt{x^{2} + 2}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 0} x^{2} + 2}$

Step 1.2

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

$\sqrt{lim_{x \to 0} x^{2} + lim_{x \to 0} 2}$

Step 1.3

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

$\sqrt{{(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 2}$

Step 1.4

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

$\sqrt{{(lim_{x \to 0} x)}^{2} + 2}$

Step 2

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

$\sqrt{0^{2} + 2}$

Step 3

Simplify the answer.

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Step 3.1

Raising $0$ to any positive power yields $0$ .

$\sqrt{0 + 2}$

Step 3.2

Add $0$ and $2$ .

$\sqrt{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\sqrt{2}$

Decimal Form:

$1.41421356 \dots$