Calculus Examples

$\frac{lim_{x \to 2} x (2 x^{2} - 8)}{x^{2} - 4 x - 12}$

Step 1

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

$\frac{lim_{x \to 2} x \cdot lim_{x \to 2} 2 x^{2} - 8}{x^{2} - 4 x - 12}$

Step 2

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

$\frac{lim_{x \to 2} x \cdot (lim_{x \to 2} 2 x^{2} - lim_{x \to 2} 8)}{x^{2} - 4 x - 12}$

Step 3

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

$\frac{lim_{x \to 2} x \cdot (2 lim_{x \to 2} x^{2} - lim_{x \to 2} 8)}{x^{2} - 4 x - 12}$

Step 4

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{lim_{x \to 2} x \cdot (2 {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} 8)}{x^{2} - 4 x - 12}$

Step 5

Evaluate the limit of $8$ which is constant as $x$ approaches $2$ .

$\frac{lim_{x \to 2} x \cdot (2 {(lim_{x \to 2} x)}^{2} - 1 \cdot 8)}{x^{2} - 4 x - 12}$

Step 6

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 6.1

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

$\frac{2 \cdot (2 {(lim_{x \to 2} x)}^{2} - 1 \cdot 8)}{x^{2} - 4 x - 12}$

Step 6.2

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

$\frac{2 \cdot (2 \cdot 2^{2} - 1 \cdot 8)}{x^{2} - 4 x - 12}$

Step 7

Simplify the answer.

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Step 7.1

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 7.1.1

Multiply $2$ by $2^{2}$ .

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Step 7.1.1.1

Raise $2$ to the power of $1$ .

$\frac{2 \cdot (2^{1} \cdot 2^{2} - 1 \cdot 8)}{x^{2} - 4 x - 12}$

Step 7.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \cdot (2^{1 + 2} - 1 \cdot 8)}{x^{2} - 4 x - 12}$

Step 7.1.2

Add $1$ and $2$ .

$\frac{2 \cdot (2^{3} - 1 \cdot 8)}{x^{2} - 4 x - 12}$

Step 7.2

Simplify the numerator.

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Step 7.2.1

Raise $2$ to the power of $3$ .

$\frac{2 (8 - 1 \cdot 8)}{x^{2} - 4 x - 12}$

Step 7.2.2

Multiply $- 1$ by $8$ .

$\frac{2 (8 - 8)}{x^{2} - 4 x - 12}$

Step 7.2.3

Subtract $8$ from $8$ .

$\frac{2 \cdot 0}{x^{2} - 4 x - 12}$

Step 7.3

Factor $x^{2} - 4 x - 12$ using the AC method.

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Step 7.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $- 4$ .

$- 6, 2$

Step 7.3.2

Write the factored form using these integers.

$\frac{2 \cdot 0}{(x - 6) (x + 2)}$

Step 7.4

Multiply $2$ by $0$ .

$\frac{0}{(x - 6) (x + 2)}$

Step 7.5

Divide $0$ by $(x - 6) (x + 2)$ .

$0$