Calculus Examples

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

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

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

Step 2

Evaluate the limit of $1$ which is constant as $x$ approaches $- 8$ .

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

Step 3

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

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

Step 4

Evaluate the limits by plugging in $- 8$ for all occurrences of $x$ .

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Step 4.1

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

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

Step 4.2

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

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

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 8$ to the power of $2$ .

$\frac{1 + 8 - 1 \cdot 64}{3 x^{2} - 8}$

Step 5.1.2

Multiply $- 1$ by $64$ .

$\frac{1 + 8 - 64}{3 x^{2} - 8}$

Step 5.1.3

Add $1$ and $8$ .

$\frac{9 - 64}{3 x^{2} - 8}$

Step 5.1.4

Subtract $64$ from $9$ .

$\frac{- 55}{3 x^{2} - 8}$

Step 5.2

Move the negative in front of the fraction.

$- \frac{55}{3 x^{2} - 8}$