Calculus Examples

$\frac{lim_{x \to 8} (- 6 x^{2}) + 6 x - 2}{- 5 x^{2} + 6}$

Step 1

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

$\frac{- lim_{x \to 8} 6 x^{2} + lim_{x \to 8} 6 x - lim_{x \to 8} 2}{- 5 x^{2} + 6}$

Step 2

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 6 lim_{x \to 8} x^{2} + lim_{x \to 8} 6 x - lim_{x \to 8} 2}{- 5 x^{2} + 6}$

Step 3

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

$\frac{- 6 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 6 x - lim_{x \to 8} 2}{- 5 x^{2} + 6}$

Step 4

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 6 {(lim_{x \to 8} x)}^{2} + 6 lim_{x \to 8} x - lim_{x \to 8} 2}{- 5 x^{2} + 6}$

Step 5

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

$\frac{- 6 {(lim_{x \to 8} x)}^{2} + 6 lim_{x \to 8} x - 1 \cdot 2}{- 5 x^{2} + 6}$

Step 6

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

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Step 6.1

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

$\frac{- 6 \cdot 8^{2} + 6 lim_{x \to 8} x - 1 \cdot 2}{- 5 x^{2} + 6}$

Step 6.2

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

$\frac{- 6 \cdot 8^{2} + 6 \cdot 8 - 1 \cdot 2}{- 5 x^{2} + 6}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$\frac{- 6 \cdot 64 + 6 \cdot 8 - 1 \cdot 2}{- 5 x^{2} + 6}$

Step 7.1.2

Multiply $- 6$ by $64$ .

$\frac{- 384 + 6 \cdot 8 - 1 \cdot 2}{- 5 x^{2} + 6}$

Step 7.1.3

Multiply $6$ by $8$ .

$\frac{- 384 + 48 - 1 \cdot 2}{- 5 x^{2} + 6}$

Step 7.1.4

Multiply $- 1$ by $2$ .

$\frac{- 384 + 48 - 2}{- 5 x^{2} + 6}$

Step 7.1.5

Add $- 384$ and $48$ .

$\frac{- 336 - 2}{- 5 x^{2} + 6}$

Step 7.1.6

Subtract $2$ from $- 336$ .

$\frac{- 338}{- 5 x^{2} + 6}$

Step 7.2

Move the negative in front of the fraction.

$- \frac{338}{- 5 x^{2} + 6}$

Step 7.3

Factor $- 1$ out of $- 5 x^{2}$ .

$- \frac{338}{- (5 x^{2}) + 6}$

Step 7.4

Rewrite $6$ as $- 1 (- 6)$ .

$- \frac{338}{- (5 x^{2}) - 1 (- 6)}$

Step 7.5

Factor $- 1$ out of $- (5 x^{2}) - 1 (- 6)$ .

$- \frac{338}{- (5 x^{2} - 6)}$

Step 7.6

Rewrite $- (5 x^{2} - 6)$ as $- 1 (5 x^{2} - 6)$ .

$- \frac{338}{- 1 (5 x^{2} - 6)}$

Step 7.7

Move the negative in front of the fraction.

$- - \frac{338}{5 x^{2} - 6}$

Step 7.8

Multiply $- 1$ by $- 1$ .

$1 \frac{338}{5 x^{2} - 6}$

Step 7.9

Multiply $\frac{338}{5 x^{2} - 6}$ by $1$ .

$\frac{338}{5 x^{2} - 6}$