Calculus Examples

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

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 $1$ .

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

Step 1.2

Move the limit under the radical sign.

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

Step 1.3

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

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

Step 2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Any root of $1$ is $1$ .

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

Step 3.1.2

Multiply $- 1$ by $1$ .

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

Step 3.1.3

Subtract $1$ from $1$ .

$\frac{0}{x - 1}$

Step 3.2

Divide $0$ by $x - 1$ .

$0$