Calculus Examples

$\frac{lim_{x \to 2} | x - 3 | - 1}{x^{3} - 8}$

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 $2$ .

$\frac{lim_{x \to 2} | x - 3 | - lim_{x \to 2} 1}{x^{3} - 8}$

Step 1.2

Move the limit inside the absolute value signs.

$\frac{| lim_{x \to 2} x - 3 | - lim_{x \to 2} 1}{x^{3} - 8}$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{| lim_{x \to 2} x - lim_{x \to 2} 3 | - lim_{x \to 2} 1}{x^{3} - 8}$

Step 1.4

Evaluate the limit of $3$ which is constant as $x$ approaches $2$ .

$\frac{| lim_{x \to 2} x - 1 \cdot 3 | - lim_{x \to 2} 1}{x^{3} - 8}$

Step 1.5

Evaluate the limit of $1$ which is constant as $x$ approaches $2$ .

$\frac{| lim_{x \to 2} x - 1 \cdot 3 | - 1 \cdot 1}{x^{3} - 8}$

Step 2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{| 2 - 1 \cdot 3 | - 1 \cdot 1}{x^{3} - 8}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $- 1$ by $3$ .

$\frac{| 2 - 3 | - 1 \cdot 1}{x^{3} - 8}$

Step 3.1.2

Subtract $3$ from $2$ .

$\frac{| - 1 | - 1 \cdot 1}{x^{3} - 8}$

Step 3.1.3

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

$\frac{1 - 1 \cdot 1}{x^{3} - 8}$

Step 3.1.4

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{x^{3} - 8}$

Step 3.1.5

Subtract $1$ from $1$ .

$\frac{0}{x^{3} - 8}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Rewrite $8$ as $2^{3}$ .

$\frac{0}{x^{3} - 2^{3}}$

Step 3.2.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 2$ .

$\frac{0}{(x - 2) (x^{2} + x \cdot 2 + 2^{2})}$

Step 3.2.3

Simplify.

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Step 3.2.3.1

Move $2$ to the left of $x$ .

$\frac{0}{(x - 2) (x^{2} + 2 \cdot x + 2^{2})}$

Step 3.2.3.2

Raise $2$ to the power of $2$ .

$\frac{0}{(x - 2) (x^{2} + 2 x + 4)}$

Step 3.3

Divide $0$ by $(x - 2) (x^{2} + 2 x + 4)$ .

$0$