Calculus Examples

$\frac{lim_{x \to - 4} \sqrt{x^{2} + 9} - 5}{x + 4}$

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 4$ .

$\frac{lim_{x \to - 4} \sqrt{x^{2} + 9} - lim_{x \to - 4} 5}{x + 4}$

Step 1.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to - 4} x^{2} + 9} - lim_{x \to - 4} 5}{x + 4}$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 4$ .

$\frac{\sqrt{lim_{x \to - 4} x^{2} + lim_{x \to - 4} 9} - lim_{x \to - 4} 5}{x + 4}$

Step 1.4

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

$\frac{\sqrt{{(lim_{x \to - 4} x)}^{2} + lim_{x \to - 4} 9} - lim_{x \to - 4} 5}{x + 4}$

Step 1.5

Evaluate the limit of $9$ which is constant as $x$ approaches $- 4$ .

$\frac{\sqrt{{(lim_{x \to - 4} x)}^{2} + 9} - lim_{x \to - 4} 5}{x + 4}$

Step 1.6

Evaluate the limit of $5$ which is constant as $x$ approaches $- 4$ .

$\frac{\sqrt{{(lim_{x \to - 4} x)}^{2} + 9} - 1 \cdot 5}{x + 4}$

Step 2

Evaluate the limit of $x$ by plugging in $- 4$ for $x$ .

$\frac{\sqrt{{(- 4)}^{2} + 9} - 1 \cdot 5}{x + 4}$

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

Raise $- 4$ to the power of $2$ .

$\frac{\sqrt{16 + 9} - 1 \cdot 5}{x + 4}$

Step 3.1.2

Add $16$ and $9$ .

$\frac{\sqrt{25} - 1 \cdot 5}{x + 4}$

Step 3.1.3

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{5^{2}} - 1 \cdot 5}{x + 4}$

Step 3.1.4

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

$\frac{5 - 1 \cdot 5}{x + 4}$

Step 3.1.5

Multiply $- 1$ by $5$ .

$\frac{5 - 5}{x + 4}$

Step 3.1.6

Subtract $5$ from $5$ .

$\frac{0}{x + 4}$

Step 3.2

Divide $0$ by $x + 4$ .

$0$