Calculus Examples

$\frac{lim_{x \to - 21} 20 - \sqrt{x^{2} - 41}}{x + 21}$

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 $- 21$ .

$\frac{lim_{x \to - 21} 20 - lim_{x \to - 21} \sqrt{x^{2} - 41}}{x + 21}$

Step 1.2

Evaluate the limit of $20$ which is constant as $x$ approaches $- 21$ .

$\frac{20 - lim_{x \to - 21} \sqrt{x^{2} - 41}}{x + 21}$

Step 1.3

Move the limit under the radical sign.

$\frac{20 - \sqrt{lim_{x \to - 21} x^{2} - 41}}{x + 21}$

Step 1.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 21$ .

$\frac{20 - \sqrt{lim_{x \to - 21} x^{2} - lim_{x \to - 21} 41}}{x + 21}$

Step 1.5

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

$\frac{20 - \sqrt{{(lim_{x \to - 21} x)}^{2} - lim_{x \to - 21} 41}}{x + 21}$

Step 1.6

Evaluate the limit of $41$ which is constant as $x$ approaches $- 21$ .

$\frac{20 - \sqrt{{(lim_{x \to - 21} x)}^{2} - 1 \cdot 41}}{x + 21}$

Step 2

Evaluate the limit of $x$ by plugging in $- 21$ for $x$ .

$\frac{20 - \sqrt{{(- 21)}^{2} - 1 \cdot 41}}{x + 21}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 21$ to the power of $2$ .

$\frac{20 - \sqrt{441 - 1 \cdot 41}}{x + 21}$

Step 3.1.2

Multiply $- 1$ by $41$ .

$\frac{20 - \sqrt{441 - 41}}{x + 21}$

Step 3.1.3

Subtract $41$ from $441$ .

$\frac{20 - \sqrt{400}}{x + 21}$

Step 3.1.4

Rewrite $400$ as $20^{2}$ .

$\frac{20 - \sqrt{20^{2}}}{x + 21}$

Step 3.1.5

Pull terms out from under the radical, assuming positive real numbers.

$\frac{20 - 1 \cdot 20}{x + 21}$

Step 3.1.6

Multiply $- 1$ by $20$ .

$\frac{20 - 20}{x + 21}$

Step 3.1.7

Subtract $20$ from $20$ .

$\frac{0}{x + 21}$

Step 3.2

Divide $0$ by $x + 21$ .

$0$