Calculus Examples

$\frac{lim_{x \to 4} (\sqrt{x + 5}) - 3}{x - 4}$

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 $4$ .

$\frac{lim_{x \to 4} \sqrt{x + 5} - lim_{x \to 4} 3}{x - 4}$

Step 1.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 4} x + 5} - lim_{x \to 4} 3}{x - 4}$

Step 1.3

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

$\frac{\sqrt{lim_{x \to 4} x + lim_{x \to 4} 5} - lim_{x \to 4} 3}{x - 4}$

Step 1.4

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

$\frac{\sqrt{lim_{x \to 4} x + 5} - lim_{x \to 4} 3}{x - 4}$

Step 1.5

Evaluate the limit of $3$ which is constant as $x$ approaches $4$ .

$\frac{\sqrt{lim_{x \to 4} x + 5} - 1 \cdot 3}{x - 4}$

Step 2

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

$\frac{\sqrt{4 + 5} - 1 \cdot 3}{x - 4}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $4$ and $5$ .

$\frac{\sqrt{9} - 1 \cdot 3}{x - 4}$

Step 3.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2}} - 1 \cdot 3}{x - 4}$

Step 3.1.3

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

$\frac{3 - 1 \cdot 3}{x - 4}$

Step 3.1.4

Multiply $- 1$ by $3$ .

$\frac{3 - 3}{x - 4}$

Step 3.1.5

Subtract $3$ from $3$ .

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

Step 3.2

Divide $0$ by $x - 4$ .

$0$