Calculus Examples

$lim_{x \to - 8} - \frac{4 {(x + 5)}^{2} + 1}{\frac{1}{3 x + 2} + 2}$

Step 1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$- lim_{x \to - 8} \frac{4 {(x + 5)}^{2} + 1}{\frac{1}{3 x + 2} + 2}$

Step 2

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

$- \frac{lim_{x \to - 8} 4 {(x + 5)}^{2} + 1}{lim_{x \to - 8} \frac{1}{3 x + 2} + 2}$

Step 3

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

$- \frac{lim_{x \to - 8} 4 {(x + 5)}^{2} + lim_{x \to - 8} 1}{lim_{x \to - 8} \frac{1}{3 x + 2} + 2}$

Step 4

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

$- \frac{4 lim_{x \to - 8} {(x + 5)}^{2} + lim_{x \to - 8} 1}{lim_{x \to - 8} \frac{1}{3 x + 2} + 2}$

Step 5

Move the exponent $2$ from ${(x + 5)}^{2}$ outside the limit using the Limits Power Rule.

$- \frac{4 {(lim_{x \to - 8} x + 5)}^{2} + lim_{x \to - 8} 1}{lim_{x \to - 8} \frac{1}{3 x + 2} + 2}$

Step 6

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

$- \frac{4 {(lim_{x \to - 8} x + lim_{x \to - 8} 5)}^{2} + lim_{x \to - 8} 1}{lim_{x \to - 8} \frac{1}{3 x + 2} + 2}$

Step 7

Evaluate the limit of $5$ which is constant as $x$ approaches $- 8$ .

$- \frac{4 {(lim_{x \to - 8} x + 5)}^{2} + lim_{x \to - 8} 1}{lim_{x \to - 8} \frac{1}{3 x + 2} + 2}$

Step 8

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

$- \frac{4 {(lim_{x \to - 8} x + 5)}^{2} + 1}{lim_{x \to - 8} \frac{1}{3 x + 2} + 2}$

Step 9

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

$- \frac{4 {(lim_{x \to - 8} x + 5)}^{2} + 1}{lim_{x \to - 8} \frac{1}{3 x + 2} + lim_{x \to - 8} 2}$

Step 10

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

$- \frac{4 {(lim_{x \to - 8} x + 5)}^{2} + 1}{\frac{lim_{x \to - 8} 1}{lim_{x \to - 8} 3 x + 2} + lim_{x \to - 8} 2}$

Step 11

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

$- \frac{4 {(lim_{x \to - 8} x + 5)}^{2} + 1}{\frac{1}{lim_{x \to - 8} 3 x + 2} + lim_{x \to - 8} 2}$

Step 12

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

$- \frac{4 {(lim_{x \to - 8} x + 5)}^{2} + 1}{\frac{1}{lim_{x \to - 8} 3 x + lim_{x \to - 8} 2} + lim_{x \to - 8} 2}$

Step 13

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$- \frac{4 {(lim_{x \to - 8} x + 5)}^{2} + 1}{\frac{1}{3 lim_{x \to - 8} x + lim_{x \to - 8} 2} + lim_{x \to - 8} 2}$

Step 14

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

$- \frac{4 {(lim_{x \to - 8} x + 5)}^{2} + 1}{\frac{1}{3 lim_{x \to - 8} x + 2} + lim_{x \to - 8} 2}$

Step 15

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

$- \frac{4 {(lim_{x \to - 8} x + 5)}^{2} + 1}{\frac{1}{3 lim_{x \to - 8} x + 2} + 2}$

Step 16

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

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Step 16.1

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

$- \frac{4 {(- 8 + 5)}^{2} + 1}{\frac{1}{3 lim_{x \to - 8} x + 2} + 2}$

Step 16.2

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

$- \frac{4 {(- 8 + 5)}^{2} + 1}{\frac{1}{3 \cdot - 8 + 2} + 2}$

Step 17

Simplify the answer.

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Step 17.1

Multiply the numerator and denominator of the fraction by $3 \cdot - 8 + 2$ .

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Step 17.1.1

Multiply $\frac{4 {(- 8 + 5)}^{2} + 1}{\frac{1}{3 \cdot - 8 + 2} + 2}$ by $\frac{3 \cdot - 8 + 2}{3 \cdot - 8 + 2}$ .

$- (\frac{3 \cdot - 8 + 2}{3 \cdot - 8 + 2} \cdot \frac{4 {(- 8 + 5)}^{2} + 1}{\frac{1}{3 \cdot - 8 + 2} + 2})$

Step 17.1.2

Combine.

$- \frac{(3 \cdot - 8 + 2) (4 {(- 8 + 5)}^{2} + 1)}{(3 \cdot - 8 + 2) (\frac{1}{3 \cdot - 8 + 2} + 2)}$

Step 17.2

Apply the distributive property.

$- \frac{(3 \cdot - 8 + 2) (4 {(- 8 + 5)}^{2}) + (3 \cdot - 8 + 2) \cdot 1}{(3 \cdot - 8 + 2) \frac{1}{3 \cdot - 8 + 2} + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.3

Simplify the denominator.

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Step 17.3.1

Multiply $3$ by $- 8$ .

$- \frac{(3 \cdot - 8 + 2) (4 {(- 8 + 5)}^{2}) + (3 \cdot - 8 + 2) \cdot 1}{(3 \cdot - 8 + 2) \frac{1}{- 24 + 2} + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.3.2

Add $- 24$ and $2$ .

$- \frac{(3 \cdot - 8 + 2) (4 {(- 8 + 5)}^{2}) + (3 \cdot - 8 + 2) \cdot 1}{(3 \cdot - 8 + 2) \frac{1}{- 22} + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.4

Simplify the numerator.

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Step 17.4.1

Multiply $3$ by $- 8$ .

$- \frac{(- 24 + 2) \cdot 4 {(- 8 + 5)}^{2} + (3 \cdot - 8 + 2) \cdot 1}{(3 \cdot - 8 + 2) \frac{1}{- 22} + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.4.2

Add $- 24$ and $2$ .

$- \frac{- 22 \cdot 4 {(- 8 + 5)}^{2} + (3 \cdot - 8 + 2) \cdot 1}{(3 \cdot - 8 + 2) \frac{1}{- 22} + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.4.3

Multiply $- 22$ by $4$ .

$- \frac{- 88 {(- 8 + 5)}^{2} + (3 \cdot - 8 + 2) \cdot 1}{(3 \cdot - 8 + 2) \frac{1}{- 22} + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.4.4

Add $- 8$ and $5$ .

$- \frac{- 88 {(- 3)}^{2} + (3 \cdot - 8 + 2) \cdot 1}{(3 \cdot - 8 + 2) \frac{1}{- 22} + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.4.5

Raise $- 3$ to the power of $2$ .

$- \frac{- 88 \cdot 9 + (3 \cdot - 8 + 2) \cdot 1}{(3 \cdot - 8 + 2) \frac{1}{- 22} + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.4.6

Multiply $- 88$ by $9$ .

$- \frac{- 792 + (3 \cdot - 8 + 2) \cdot 1}{(3 \cdot - 8 + 2) \frac{1}{- 22} + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.4.7

Multiply $3$ by $- 8$ .

$- \frac{- 792 + (- 24 + 2) \cdot 1}{(3 \cdot - 8 + 2) \frac{1}{- 22} + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.4.8

Add $- 24$ and $2$ .

$- \frac{- 792 - 22 \cdot 1}{(3 \cdot - 8 + 2) \frac{1}{- 22} + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.4.9

Multiply $- 22$ by $1$ .

$- \frac{- 792 - 22}{(3 \cdot - 8 + 2) \frac{1}{- 22} + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.4.10

Subtract $22$ from $- 792$ .

$- \frac{- 814}{(3 \cdot - 8 + 2) \frac{1}{- 22} + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.5

Simplify the denominator.

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Step 17.5.1

Multiply $3$ by $- 8$ .

$- \frac{- 814}{(- 24 + 2) \frac{1}{- 22} + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.5.2

Add $- 24$ and $2$ .

$- \frac{- 814}{- 22 (\frac{1}{- 22}) + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.5.3

Cancel the common factor of $22$ .

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Step 17.5.3.1

Factor $22$ out of $- 22$ .

$- \frac{- 814}{22 (- 1) \frac{1}{- 22} + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.5.3.2

Factor $22$ out of $- 22$ .

$- \frac{- 814}{22 \cdot - 1 \frac{1}{22 \cdot - 1} + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.5.3.3

Cancel the common factor.

$- \frac{- 814}{22 \cdot - 1 \frac{1}{22 \cdot - 1} + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.5.3.4

Rewrite the expression.

$- \frac{- 814}{- 1 (\frac{1}{- 1}) + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.5.4

Divide $1$ by $- 1$ .

$- \frac{- 814}{- 1 \cdot - 1 + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.5.5

Multiply $- 1$ by $- 1$ .

$- \frac{- 814}{1 + (3 \cdot - 8 + 2) \cdot 2}$

Step 17.5.6

Multiply $3$ by $- 8$ .

$- \frac{- 814}{1 + (- 24 + 2) \cdot 2}$

Step 17.5.7

Add $- 24$ and $2$ .

$- \frac{- 814}{1 - 22 \cdot 2}$

Step 17.5.8

Multiply $- 22$ by $2$ .

$- \frac{- 814}{1 - 44}$

Step 17.5.9

Subtract $44$ from $1$ .

$- \frac{- 814}{- 43}$

Step 17.6

Dividing two negative values results in a positive value.

$- \frac{814}{43}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$- \frac{814}{43}$

Decimal Form:

$- 18.93023255 \dots$