Calculus Examples

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

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 $2$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Multiply the numerator and denominator of the fraction by $2$ .

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Step 3.1.1

Multiply $\frac{\frac{1}{2} - \frac{1}{2}}{x^{3} - 8}$ by $\frac{2}{2}$ .

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

Step 3.1.2

Combine.

$\frac{2 (\frac{1}{2} - \frac{1}{2})}{2 (x^{3} - 8)}$

Step 3.2

Apply the distributive property.

$\frac{2 (\frac{1}{2}) + 2 (- \frac{1}{2})}{2 x^{3} + 2 \cdot - 8}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $2$ .

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Step 3.3.1.1

Cancel the common factor.

$\frac{2 (\frac{1}{2}) + 2 (- \frac{1}{2})}{2 x^{3} + 2 \cdot - 8}$

Step 3.3.1.2

Rewrite the expression.

$\frac{1 + 2 (- \frac{1}{2})}{2 x^{3} + 2 \cdot - 8}$

Step 3.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\frac{1 + 2 (\frac{- 1}{2})}{2 x^{3} + 2 \cdot - 8}$

Step 3.3.2.2

Cancel the common factor.

$\frac{1 + 2 (\frac{- 1}{2})}{2 x^{3} + 2 \cdot - 8}$

Step 3.3.2.3

Rewrite the expression.

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

Step 3.4

Subtract $1$ from $1$ .

$\frac{0}{2 x^{3} + 2 \cdot - 8}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Factor $2$ out of $2 x^{3} + 2 \cdot - 8$ .

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Step 3.5.1.1

Factor $2$ out of $2 x^{3}$ .

$\frac{0}{2 (x^{3}) + 2 \cdot - 8}$

Step 3.5.1.2

Factor $2$ out of $2 \cdot - 8$ .

$\frac{0}{2 (x^{3}) + 2 (- 8)}$

Step 3.5.1.3

Factor $2$ out of $2 (x^{3}) + 2 (- 8)$ .

$\frac{0}{2 (x^{3} - 8)}$

Step 3.5.2

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

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

Step 3.5.3

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 $b = 2$ .

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

Step 3.5.4

Simplify.

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Step 3.5.4.1

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

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

Step 3.5.4.2

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

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

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

Step 3.6

Cancel the common factor of $0$ and $2$ .

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Step 3.6.1

Factor $2$ out of $0$ .

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

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $2$ out of $2 (x - 2) (x^{2} + 2 x + 4)$ .

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

Step 3.6.2.2

Cancel the common factor.

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

Step 3.6.2.3

Rewrite the expression.

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

Step 3.7

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

$0$