Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Move the limit inside the absolute value signs.

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

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{2}$ .

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

Step 1.3

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

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

Step 1.4

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{1}{2}$ .

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

Step 2

Evaluate the limit of $x$ by plugging in $\frac{1}{2}$ for $x$ .

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Cancel the common factor of $2$ .

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Step 3.1.1.1

Cancel the common factor.

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

Step 3.1.1.2

Rewrite the expression.

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

Step 3.1.2

Multiply $- 1$ by $1$ .

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

Step 3.1.3

Subtract $1$ from $1$ .

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

Step 3.1.4

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$\frac{0}{2 x - 1}$

Step 3.2

Divide $0$ by $2 x - 1$ .

$0$