Calculus Examples

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

Step 1

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

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

Step 2

Move the limit under the radical sign.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 6.1

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

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

Step 6.2

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

$\frac{\sqrt{2 \cdot 1} \cdot (1 - 1 \cdot 1)}{| x - 1 |}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Multiply $- 1$ by $1$ .

$\frac{\sqrt{2 \cdot 1} (1 - 1)}{| x - 1 |}$

Step 7.1.2

Subtract $1$ from $1$ .

$\frac{\sqrt{2 \cdot 1} \cdot 0}{| x - 1 |}$

Step 7.1.3

Multiply $2$ by $1$ .

$\frac{\sqrt{2} \cdot 0}{| x - 1 |}$

Step 7.2

Multiply $\sqrt{2}$ by $0$ .

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

Step 7.3

Divide $0$ by $| x - 1 |$ .

$0$