Calculus Examples

$\frac{lim_{x \to 1} 1 - \sqrt{2 x^{2} - 1}}{x - 1}$

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 $1$ .

$\frac{lim_{x \to 1} 1 - lim_{x \to 1} \sqrt{2 x^{2} - 1}}{x - 1}$

Step 1.2

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

$\frac{1 - lim_{x \to 1} \sqrt{2 x^{2} - 1}}{x - 1}$

Step 1.3

Move the limit under the radical sign.

$\frac{1 - \sqrt{lim_{x \to 1} 2 x^{2} - 1}}{x - 1}$

Step 1.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{1 - \sqrt{lim_{x \to 1} 2 x^{2} - lim_{x \to 1} 1}}{x - 1}$

Step 1.5

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{1 - \sqrt{2 lim_{x \to 1} x^{2} - lim_{x \to 1} 1}}{x - 1}$

Step 1.6

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{1 - \sqrt{2 {(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 1}}{x - 1}$

Step 1.7

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

$\frac{1 - \sqrt{2 {(lim_{x \to 1} x)}^{2} - 1 \cdot 1}}{x - 1}$

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

One to any power is one.

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

Step 3.1.2

Multiply $2$ by $1$ .

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

Step 3.1.3

Multiply $- 1$ by $1$ .

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

Step 3.1.4

Subtract $1$ from $2$ .

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

Step 3.1.5

Any root of $1$ is $1$ .

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

Step 3.1.6

Multiply $- 1$ by $1$ .

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

Step 3.1.7

Subtract $1$ from $1$ .

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

Step 3.2

Divide $0$ by $x - 1$ .

$0$