Calculus Examples

$\frac{lim_{x \to - 2} 4 x^{2 + 6 x - 4}}{2 x^{2} - 8}$

Step 1

Subtract $4$ from $2$ .

$\frac{lim_{x \to - 2} 4 x^{6 x - 2}}{2 x^{2} - 8}$

Step 2

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

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Step 2.1

Evaluate the limit of $4 x^{6 x - 2}$ by plugging in $- 2$ for $x$ .

$\frac{4 {(- 2)}^{6 \cdot - 2 - 2}}{2 x^{2} - 8}$

Step 2.2

Multiply $6$ by $- 2$ .

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

Step 2.3

Subtract $2$ from $- 12$ .

$\frac{4 {(- 2)}^{- 14}}{2 x^{2} - 8}$

Step 2.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4 \frac{1}{{(- 2)}^{14}}}{2 x^{2} - 8}$

Step 2.5

Raise $- 2$ to the power of $14$ .

$\frac{4 (\frac{1}{16384})}{2 x^{2} - 8}$

Step 2.6

Cancel the common factor of $4$ .

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Step 2.6.1

Factor $4$ out of $16384$ .

$\frac{4 \frac{1}{4 (4096)}}{2 x^{2} - 8}$

Step 2.6.2

Cancel the common factor.

$\frac{4 \frac{1}{4 \cdot 4096}}{2 x^{2} - 8}$

Step 2.6.3

Rewrite the expression.

$\frac{\frac{1}{4096}}{2 x^{2} - 8}$

Step 3

Simplify the answer.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{4096} \cdot \frac{1}{2 x^{2} - 8}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Factor $2$ out of $2 x^{2} - 8$ .

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Step 3.2.1.1

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

$\frac{1}{4096} \cdot \frac{1}{2 (x^{2}) - 8}$

Step 3.2.1.2

Factor $2$ out of $- 8$ .

$\frac{1}{4096} \cdot \frac{1}{2 x^{2} + 2 \cdot - 4}$

Step 3.2.1.3

Factor $2$ out of $2 x^{2} + 2 \cdot - 4$ .

$\frac{1}{4096} \cdot \frac{1}{2 (x^{2} - 4)}$

Step 3.2.2

Rewrite $4$ as $2^{2}$ .

$\frac{1}{4096} \cdot \frac{1}{2 (x^{2} - 2^{2})}$

Step 3.2.3

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

$\frac{1}{4096} \cdot \frac{1}{2 (x + 2) (x - 2)}$

Step 3.3

Multiply $\frac{1}{4096} \cdot \frac{1}{2 (x + 2) (x - 2)}$ .

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Step 3.3.1

Multiply $\frac{1}{4096}$ by $\frac{1}{2 (x + 2) (x - 2)}$ .

$\frac{1}{4096 (2 (x + 2) (x - 2))}$

Step 3.3.2

Multiply $2$ by $4096$ .

$\frac{1}{8192 ((x + 2) (x - 2))}$

$\frac{1}{8192 (x + 2) (x - 2)}$