Calculus Examples

$\frac{lim_{x \to 0} \sqrt{2 - x^{2}} - \sqrt{2 + x^{2}}}{5 x^{2}}$

Step 1

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

$\frac{lim_{x \to 0} \sqrt{2 - x^{2}} - lim_{x \to 0} \sqrt{2 + x^{2}}}{5 x^{2}}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 0} 2 - x^{2}} - lim_{x \to 0} \sqrt{2 + x^{2}}}{5 x^{2}}$

Step 3

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

$\frac{\sqrt{lim_{x \to 0} 2 - lim_{x \to 0} x^{2}} - lim_{x \to 0} \sqrt{2 + x^{2}}}{5 x^{2}}$

Step 4

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

$\frac{\sqrt{2 - lim_{x \to 0} x^{2}} - lim_{x \to 0} \sqrt{2 + x^{2}}}{5 x^{2}}$

Step 5

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

$\frac{\sqrt{2 - {(lim_{x \to 0} x)}^{2}} - lim_{x \to 0} \sqrt{2 + x^{2}}}{5 x^{2}}$

Step 6

Move the limit under the radical sign.

$\frac{\sqrt{2 - {(lim_{x \to 0} x)}^{2}} - \sqrt{lim_{x \to 0} 2 + x^{2}}}{5 x^{2}}$

Step 7

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

$\frac{\sqrt{2 - {(lim_{x \to 0} x)}^{2}} - \sqrt{lim_{x \to 0} 2 + lim_{x \to 0} x^{2}}}{5 x^{2}}$

Step 8

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

$\frac{\sqrt{2 - {(lim_{x \to 0} x)}^{2}} - \sqrt{2 + lim_{x \to 0} x^{2}}}{5 x^{2}}$

Step 9

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

$\frac{\sqrt{2 - {(lim_{x \to 0} x)}^{2}} - \sqrt{2 + {(lim_{x \to 0} x)}^{2}}}{5 x^{2}}$

Step 10

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 10.1

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

$\frac{\sqrt{2 - 0^{2}} - \sqrt{2 + {(lim_{x \to 0} x)}^{2}}}{5 x^{2}}$

Step 10.2

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

$\frac{\sqrt{2 - 0^{2}} - \sqrt{2 + 0^{2}}}{5 x^{2}}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

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

$\frac{\sqrt{2 - 0} - \sqrt{2 + 0^{2}}}{5 x^{2}}$

Step 11.1.2

Multiply $- 1$ by $0$ .

$\frac{\sqrt{2 + 0} - \sqrt{2 + 0^{2}}}{5 x^{2}}$

Step 11.1.3

Add $2$ and $0$ .

$\frac{\sqrt{2} - \sqrt{2 + 0^{2}}}{5 x^{2}}$

Step 11.1.4

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

$\frac{\sqrt{2} - \sqrt{2 + 0}}{5 x^{2}}$

Step 11.1.5

Add $2$ and $0$ .

$\frac{\sqrt{2} - \sqrt{2}}{5 x^{2}}$

Step 11.1.6

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

$\frac{0}{5 x^{2}}$

Step 11.2

Cancel the common factor of $0$ and $5$ .

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Step 11.2.1

Factor $5$ out of $0$ .

$\frac{5 (0)}{5 x^{2}}$

Step 11.2.2

Cancel the common factors.

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Step 11.2.2.1

Factor $5$ out of $5 x^{2}$ .

$\frac{5 (0)}{5 (x^{2})}$

Step 11.2.2.2

Cancel the common factor.

$\frac{5 \cdot 0}{5 x^{2}}$

Step 11.2.2.3

Rewrite the expression.

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

Step 11.3

Divide $0$ by $x^{2}$ .

$0$