Calculus Examples

$lim_{x \to - 4} \frac{4 x^{2} + 4 x - 8}{a x^{2} + b x + 8} = \frac{9}{5}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 4$ .

$\frac{lim_{x \to - 4} 4 x^{2} + 4 x - 8}{lim_{x \to - 4} a x^{2} + b x + 8} = \frac{9}{5}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 4$ .

$\frac{lim_{x \to - 4} 4 x^{2} + lim_{x \to - 4} 4 x - lim_{x \to - 4} 8}{lim_{x \to - 4} a x^{2} + b x + 8} = \frac{9}{5}$

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 lim_{x \to - 4} x^{2} + lim_{x \to - 4} 4 x - lim_{x \to - 4} 8}{lim_{x \to - 4} a x^{2} + b x + 8} = \frac{9}{5}$

Step 4

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

$\frac{4 {(lim_{x \to - 4} x)}^{2} + lim_{x \to - 4} 4 x - lim_{x \to - 4} 8}{lim_{x \to - 4} a x^{2} + b x + 8} = \frac{9}{5}$

Step 5

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 {(lim_{x \to - 4} x)}^{2} + 4 lim_{x \to - 4} x - lim_{x \to - 4} 8}{lim_{x \to - 4} a x^{2} + b x + 8} = \frac{9}{5}$

Step 6

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

$\frac{4 {(lim_{x \to - 4} x)}^{2} + 4 lim_{x \to - 4} x - 1 \cdot 8}{lim_{x \to - 4} a x^{2} + b x + 8} = \frac{9}{5}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 4$ .

$\frac{4 {(lim_{x \to - 4} x)}^{2} + 4 lim_{x \to - 4} x - 1 \cdot 8}{lim_{x \to - 4} a x^{2} + lim_{x \to - 4} b x + lim_{x \to - 4} 8} = \frac{9}{5}$

Step 8

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

$\frac{4 {(lim_{x \to - 4} x)}^{2} + 4 lim_{x \to - 4} x - 1 \cdot 8}{a lim_{x \to - 4} x^{2} + lim_{x \to - 4} b x + lim_{x \to - 4} 8} = \frac{9}{5}$

Step 9

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

$\frac{4 {(lim_{x \to - 4} x)}^{2} + 4 lim_{x \to - 4} x - 1 \cdot 8}{a {(lim_{x \to - 4} x)}^{2} + lim_{x \to - 4} b x + lim_{x \to - 4} 8} = \frac{9}{5}$

Step 10

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

$\frac{4 {(lim_{x \to - 4} x)}^{2} + 4 lim_{x \to - 4} x - 1 \cdot 8}{a {(lim_{x \to - 4} x)}^{2} + b lim_{x \to - 4} x + lim_{x \to - 4} 8} = \frac{9}{5}$

Step 11

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

$\frac{4 {(lim_{x \to - 4} x)}^{2} + 4 lim_{x \to - 4} x - 1 \cdot 8}{a {(lim_{x \to - 4} x)}^{2} + b lim_{x \to - 4} x + 8} = \frac{9}{5}$

Step 12

Evaluate the limits by plugging in $- 4$ for all occurrences of $x$ .

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Step 12.1

Evaluate the limit of $x$ by plugging in $- 4$ for $x$ .

$\frac{4 {(- 4)}^{2} + 4 lim_{x \to - 4} x - 1 \cdot 8}{a {(lim_{x \to - 4} x)}^{2} + b lim_{x \to - 4} x + 8} = \frac{9}{5}$

Step 12.2

Evaluate the limit of $x$ by plugging in $- 4$ for $x$ .

$\frac{4 {(- 4)}^{2} + 4 \cdot - 4 - 1 \cdot 8}{a {(lim_{x \to - 4} x)}^{2} + b lim_{x \to - 4} x + 8} = \frac{9}{5}$

Step 12.3

Evaluate the limit of $x$ by plugging in $- 4$ for $x$ .

$\frac{4 {(- 4)}^{2} + 4 \cdot - 4 - 1 \cdot 8}{a {(- 4)}^{2} + b lim_{x \to - 4} x + 8} = \frac{9}{5}$

Step 12.4

Evaluate the limit of $x$ by plugging in $- 4$ for $x$ .

$\frac{4 {(- 4)}^{2} + 4 \cdot - 4 - 1 \cdot 8}{a {(- 4)}^{2} + b \cdot - 4 + 8} = \frac{9}{5}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Raise $- 4$ to the power of $2$ .

$\frac{4 \cdot 16 + 4 \cdot - 4 - 1 \cdot 8}{a {(- 4)}^{2} + b \cdot - 4 + 8} = \frac{9}{5}$

Step 13.1.2

Multiply $4$ by $16$ .

$\frac{64 + 4 \cdot - 4 - 1 \cdot 8}{a {(- 4)}^{2} + b \cdot - 4 + 8} = \frac{9}{5}$

Step 13.1.3

Multiply $4$ by $- 4$ .

$\frac{64 - 16 - 1 \cdot 8}{a {(- 4)}^{2} + b \cdot - 4 + 8} = \frac{9}{5}$

Step 13.1.4

Multiply $- 1$ by $8$ .

$\frac{64 - 16 - 8}{a {(- 4)}^{2} + b \cdot - 4 + 8} = \frac{9}{5}$

Step 13.1.5

Subtract $16$ from $64$ .

$\frac{48 - 8}{a {(- 4)}^{2} + b \cdot - 4 + 8} = \frac{9}{5}$

Step 13.1.6

Subtract $8$ from $48$ .

$\frac{40}{a {(- 4)}^{2} + b \cdot - 4 + 8} = \frac{9}{5}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Raise $- 4$ to the power of $2$ .

$\frac{40}{a \cdot 16 + b \cdot - 4 + 8} = \frac{9}{5}$

Step 13.2.2

Move $16$ to the left of $a$ .

$\frac{40}{16 \cdot a + b \cdot - 4 + 8} = \frac{9}{5}$

Step 13.2.3

Move $- 4$ to the left of $b$ .

$\frac{40}{16 a - 4 \cdot b + 8} = \frac{9}{5}$

Step 13.2.4

Factor $4$ out of $16 a - 4 b + 8$ .

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Step 13.2.4.1

Factor $4$ out of $16 a$ .

$\frac{40}{4 (4 a) - 4 b + 8} = \frac{9}{5}$

Step 13.2.4.2

Factor $4$ out of $- 4 b$ .

$\frac{40}{4 (4 a) + 4 (- b) + 8} = \frac{9}{5}$

Step 13.2.4.3

Factor $4$ out of $8$ .

$\frac{40}{4 (4 a) + 4 (- b) + 4 (2)} = \frac{9}{5}$

Step 13.2.4.4

Factor $4$ out of $4 (4 a) + 4 (- b)$ .

$\frac{40}{4 (4 a - b) + 4 (2)} = \frac{9}{5}$

Step 13.2.4.5

Factor $4$ out of $4 (4 a - b) + 4 (2)$ .

$\frac{40}{4 (4 a - b + 2)} = \frac{9}{5}$

Step 13.3

Cancel the common factor of $40$ and $4$ .

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Step 13.3.1

Factor $4$ out of $40$ .

$\frac{4 \cdot 10}{4 (4 a - b + 2)} = \frac{9}{5}$

Step 13.3.2

Cancel the common factors.

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Step 13.3.2.1

Cancel the common factor.

$\frac{4 \cdot 10}{4 (4 a - b + 2)} = \frac{9}{5}$

Step 13.3.2.2

Rewrite the expression.

$\frac{10}{4 a - b + 2} = \frac{9}{5}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{10}{4 a - b + 2} = \frac{9}{5}$

Decimal Form:

$\frac{10}{4 a - b + 2} = 1.8$