Calculus Examples

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

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 $4$ .

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

Step 1.2

Move the limit under the radical sign.

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Evaluate the limit of $1$ which is constant as $x$ approaches $4$ .

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $- 1$ by $3$ .

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

Step 3.1.2

Subtract $3$ from $4$ .

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

Step 3.1.3

Any root of $1$ is $1$ .

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

Step 3.1.4

Multiply $- 1$ by $1$ .

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

Step 3.1.5

Subtract $1$ from $1$ .

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

Step 3.2

Divide $0$ by $x - 4$ .

$0$