Calculus Examples

$lim_{x \to - 8} \frac{5 x^{2} - 6 x - 4}{10 - 2 x - 5 x^{2}}$

Step 1

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

$\frac{lim_{x \to - 8} 5 x^{2} - 6 x - 4}{lim_{x \to - 8} 10 - 2 x - 5 x^{2}}$

Step 2

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

$\frac{lim_{x \to - 8} 5 x^{2} - lim_{x \to - 8} 6 x - lim_{x \to - 8} 4}{lim_{x \to - 8} 10 - 2 x - 5 x^{2}}$

Step 3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 lim_{x \to - 8} x^{2} - lim_{x \to - 8} 6 x - lim_{x \to - 8} 4}{lim_{x \to - 8} 10 - 2 x - 5 x^{2}}$

Step 4

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

$\frac{5 {(lim_{x \to - 8} x)}^{2} - lim_{x \to - 8} 6 x - lim_{x \to - 8} 4}{lim_{x \to - 8} 10 - 2 x - 5 x^{2}}$

Step 5

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

$\frac{5 {(lim_{x \to - 8} x)}^{2} - 6 lim_{x \to - 8} x - lim_{x \to - 8} 4}{lim_{x \to - 8} 10 - 2 x - 5 x^{2}}$

Step 6

Evaluate the limit of $4$ which is constant as $x$ approaches $- 8$ .

$\frac{5 {(lim_{x \to - 8} x)}^{2} - 6 lim_{x \to - 8} x - 1 \cdot 4}{lim_{x \to - 8} 10 - 2 x - 5 x^{2}}$

Step 7

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

$\frac{5 {(lim_{x \to - 8} x)}^{2} - 6 lim_{x \to - 8} x - 1 \cdot 4}{lim_{x \to - 8} 10 - lim_{x \to - 8} 2 x - lim_{x \to - 8} 5 x^{2}}$

Step 8

Evaluate the limit of $10$ which is constant as $x$ approaches $- 8$ .

$\frac{5 {(lim_{x \to - 8} x)}^{2} - 6 lim_{x \to - 8} x - 1 \cdot 4}{10 - lim_{x \to - 8} 2 x - lim_{x \to - 8} 5 x^{2}}$

Step 9

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

$\frac{5 {(lim_{x \to - 8} x)}^{2} - 6 lim_{x \to - 8} x - 1 \cdot 4}{10 - 2 lim_{x \to - 8} x - lim_{x \to - 8} 5 x^{2}}$

Step 10

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 {(lim_{x \to - 8} x)}^{2} - 6 lim_{x \to - 8} x - 1 \cdot 4}{10 - 2 lim_{x \to - 8} x - 5 lim_{x \to - 8} x^{2}}$

Step 11

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

$\frac{5 {(lim_{x \to - 8} x)}^{2} - 6 lim_{x \to - 8} x - 1 \cdot 4}{10 - 2 lim_{x \to - 8} x - 5 {(lim_{x \to - 8} x)}^{2}}$

Step 12

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

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Step 12.1

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

$\frac{5 {(- 8)}^{2} - 6 lim_{x \to - 8} x - 1 \cdot 4}{10 - 2 lim_{x \to - 8} x - 5 {(lim_{x \to - 8} x)}^{2}}$

Step 12.2

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

$\frac{5 {(- 8)}^{2} - 6 \cdot - 8 - 1 \cdot 4}{10 - 2 lim_{x \to - 8} x - 5 {(lim_{x \to - 8} x)}^{2}}$

Step 12.3

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

$\frac{5 {(- 8)}^{2} - 6 \cdot - 8 - 1 \cdot 4}{10 - 2 \cdot - 8 - 5 {(lim_{x \to - 8} x)}^{2}}$

Step 12.4

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

$\frac{5 {(- 8)}^{2} - 6 \cdot - 8 - 1 \cdot 4}{10 - 2 \cdot - 8 - 5 {(- 8)}^{2}}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

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

$\frac{5 \cdot 64 - 6 \cdot - 8 - 1 \cdot 4}{10 - 2 \cdot - 8 - 5 {(- 8)}^{2}}$

Step 13.1.2

Multiply $5$ by $64$ .

$\frac{320 - 6 \cdot - 8 - 1 \cdot 4}{10 - 2 \cdot - 8 - 5 {(- 8)}^{2}}$

Step 13.1.3

Multiply $- 6$ by $- 8$ .

$\frac{320 + 48 - 1 \cdot 4}{10 - 2 \cdot - 8 - 5 {(- 8)}^{2}}$

Step 13.1.4

Multiply $- 1$ by $4$ .

$\frac{320 + 48 - 4}{10 - 2 \cdot - 8 - 5 {(- 8)}^{2}}$

Step 13.1.5

Add $320$ and $48$ .

$\frac{368 - 4}{10 - 2 \cdot - 8 - 5 {(- 8)}^{2}}$

Step 13.1.6

Subtract $4$ from $368$ .

$\frac{364}{10 - 2 \cdot - 8 - 5 {(- 8)}^{2}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Multiply $- 2$ by $- 8$ .

$\frac{364}{10 + 16 - 5 {(- 8)}^{2}}$

Step 13.2.2

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

$\frac{364}{10 + 16 - 5 \cdot 64}$

Step 13.2.3

Multiply $- 5$ by $64$ .

$\frac{364}{10 + 16 - 320}$

Step 13.2.4

Add $10$ and $16$ .

$\frac{364}{26 - 320}$

Step 13.2.5

Subtract $320$ from $26$ .

$\frac{364}{- 294}$

Step 13.3

Cancel the common factor of $364$ and $- 294$ .

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Step 13.3.1

Factor $14$ out of $364$ .

$\frac{14 (26)}{- 294}$

Step 13.3.2

Cancel the common factors.

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Step 13.3.2.1

Factor $14$ out of $- 294$ .

$\frac{14 \cdot 26}{14 \cdot - 21}$

Step 13.3.2.2

Cancel the common factor.

$\frac{14 \cdot 26}{14 \cdot - 21}$

Step 13.3.2.3

Rewrite the expression.

$\frac{26}{- 21}$

Step 13.4

Move the negative in front of the fraction.

$- \frac{26}{21}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$- \frac{26}{21}$

Decimal Form:

$- 1.\overline{238095}$