Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Move the limit under the radical sign.

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Add $4$ and $5$ .

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

Step 3.2

Simplify the numerator.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

To write $- \sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

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

Step 3.4

Combine $- \sqrt{3}$ and $\frac{x - 4}{x - 4}$ .

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

Step 3.5

Combine the numerators over the common denominator.

$\frac{3 - \sqrt{3} (x - 4)}{x - 4}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Apply the distributive property.

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

Step 3.6.2

Multiply $- 4$ by $- 1$ .

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

Step 3.6.3

Reorder terms.

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

Step 3.7

Factor $- 1$ out of $- x \sqrt{3}$ .

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

Step 3.8

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{- (x \sqrt{3}) - 1 (- 3) + 4 \sqrt{3}}{x - 4}$

Step 3.9

Factor $- 1$ out of $- (x \sqrt{3}) - 1 (- 3)$ .

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

Step 3.10

Factor $- 1$ out of $4 \sqrt{3}$ .

$\frac{- (x \sqrt{3} - 3) - (- 4 \sqrt{3})}{x - 4}$

Step 3.11

Factor $- 1$ out of $- (x \sqrt{3} - 3) - (- 4 \sqrt{3})$ .

$\frac{- (x \sqrt{3} - 3 - 4 \sqrt{3})}{x - 4}$

Step 3.12

Rewrite $- (x \sqrt{3} - 3 - 4 \sqrt{3})$ as $- 1 (x \sqrt{3} - 3 - 4 \sqrt{3})$ .

$\frac{- 1 (x \sqrt{3} - 3 - 4 \sqrt{3})}{x - 4}$

Step 3.13

Move the negative in front of the fraction.

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