Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Multiply $\sqrt{x + 1}$ by $1$ .

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

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

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

Step 1.3

Move the limit under the radical sign.

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

Step 1.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

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

Step 1.5

Evaluate the limit of $1$ which is constant as $x$ approaches $3$ .

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

Step 1.6

Evaluate the limit of $\sqrt{3 - 1}$ which is constant as $x$ approaches $3$ .

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

Step 2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{\sqrt{3 + 1} - \sqrt{3 - 1}}{x - 3}$

Step 3

Simplify the numerator.

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Step 3.1

Add $3$ and $1$ .

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

Step 3.2

Rewrite $4$ as $2^{2}$ .

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

Step 3.3

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

$\frac{2 - \sqrt{3 - 1}}{x - 3}$

Step 3.4

Subtract $1$ from $3$ .

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