Calculus Examples

$\frac{lim_{x \to 0} | 2 x - 1 | - | 2 x + 1 |}{x}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} | 2 x - 1 | - lim_{x \to 0} | 2 x + 1 |}{x}$

Step 2

Move the limit inside the absolute value signs.

$\frac{| lim_{x \to 0} 2 x - 1 | - lim_{x \to 0} | 2 x + 1 |}{x}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{| lim_{x \to 0} 2 x - lim_{x \to 0} 1 | - lim_{x \to 0} | 2 x + 1 |}{x}$

Step 4

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

$\frac{| 2 lim_{x \to 0} x - lim_{x \to 0} 1 | - lim_{x \to 0} | 2 x + 1 |}{x}$

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{| 2 lim_{x \to 0} x - 1 \cdot 1 | - lim_{x \to 0} | 2 x + 1 |}{x}$

Step 6

Move the limit inside the absolute value signs.

$\frac{| 2 lim_{x \to 0} x - 1 \cdot 1 | - | lim_{x \to 0} 2 x + 1 |}{x}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{| 2 lim_{x \to 0} x - 1 \cdot 1 | - | lim_{x \to 0} 2 x + lim_{x \to 0} 1 |}{x}$

Step 8

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

$\frac{| 2 lim_{x \to 0} x - 1 \cdot 1 | - | 2 lim_{x \to 0} x + lim_{x \to 0} 1 |}{x}$

Step 9

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{| 2 lim_{x \to 0} x - 1 \cdot 1 | - | 2 lim_{x \to 0} x + 1 |}{x}$

Step 10

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{| 2 \cdot 0 - 1 \cdot 1 | - | 2 lim_{x \to 0} x + 1 |}{x}$

Step 10.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Simplify each term.

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Step 11.1.1.1

Multiply $2$ by $0$ .

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

Step 11.1.1.2

Multiply $- 1$ by $1$ .

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

Step 11.1.2

Subtract $1$ from $0$ .

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

Step 11.1.3

The absolute value is the distance between a number and zero. The distance between $- 1$ and $0$ is $1$ .

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

Step 11.1.4

Multiply $2$ by $0$ .

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

Step 11.1.5

Add $0$ and $1$ .

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

Step 11.1.6

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

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

Step 11.1.7

Multiply $- 1$ by $1$ .

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

Step 11.1.8

Subtract $1$ from $1$ .

$\frac{0}{x}$

Step 11.2

Divide $0$ by $x$ .

$0$