Calculus Examples

$lim_{x \to - 8} \frac{2 x^{2} + 7}{x - 5} + \frac{3 - 4 x^{2}}{2 x + 7}$

Step 1

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

$lim_{x \to - 8} \frac{2 x^{2} + 7}{x - 5} + lim_{x \to - 8} \frac{3 - 4 x^{2}}{2 x + 7}$

Step 2

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

$\frac{lim_{x \to - 8} 2 x^{2} + 7}{lim_{x \to - 8} x - 5} + lim_{x \to - 8} \frac{3 - 4 x^{2}}{2 x + 7}$

Step 3

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

$\frac{lim_{x \to - 8} 2 x^{2} + lim_{x \to - 8} 7}{lim_{x \to - 8} x - 5} + lim_{x \to - 8} \frac{3 - 4 x^{2}}{2 x + 7}$

Step 4

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

$\frac{2 lim_{x \to - 8} x^{2} + lim_{x \to - 8} 7}{lim_{x \to - 8} x - 5} + lim_{x \to - 8} \frac{3 - 4 x^{2}}{2 x + 7}$

Step 5

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

$\frac{2 {(lim_{x \to - 8} x)}^{2} + lim_{x \to - 8} 7}{lim_{x \to - 8} x - 5} + lim_{x \to - 8} \frac{3 - 4 x^{2}}{2 x + 7}$

Step 6

Evaluate the limit of $7$ which is constant as $x$ approaches $- 8$ .

$\frac{2 {(lim_{x \to - 8} x)}^{2} + 7}{lim_{x \to - 8} x - 5} + lim_{x \to - 8} \frac{3 - 4 x^{2}}{2 x + 7}$

Step 7

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

$\frac{2 {(lim_{x \to - 8} x)}^{2} + 7}{lim_{x \to - 8} x - lim_{x \to - 8} 5} + lim_{x \to - 8} \frac{3 - 4 x^{2}}{2 x + 7}$

Step 8

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

$\frac{2 {(lim_{x \to - 8} x)}^{2} + 7}{lim_{x \to - 8} x - 1 \cdot 5} + lim_{x \to - 8} \frac{3 - 4 x^{2}}{2 x + 7}$

Step 9

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

$\frac{2 {(lim_{x \to - 8} x)}^{2} + 7}{lim_{x \to - 8} x - 1 \cdot 5} + \frac{lim_{x \to - 8} 3 - 4 x^{2}}{lim_{x \to - 8} 2 x + 7}$

Step 10

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

$\frac{2 {(lim_{x \to - 8} x)}^{2} + 7}{lim_{x \to - 8} x - 1 \cdot 5} + \frac{lim_{x \to - 8} 3 - lim_{x \to - 8} 4 x^{2}}{lim_{x \to - 8} 2 x + 7}$

Step 11

Evaluate the limit of $3$ which is constant as $x$ approaches $- 8$ .

$\frac{2 {(lim_{x \to - 8} x)}^{2} + 7}{lim_{x \to - 8} x - 1 \cdot 5} + \frac{3 - lim_{x \to - 8} 4 x^{2}}{lim_{x \to - 8} 2 x + 7}$

Step 12

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

$\frac{2 {(lim_{x \to - 8} x)}^{2} + 7}{lim_{x \to - 8} x - 1 \cdot 5} + \frac{3 - 4 lim_{x \to - 8} x^{2}}{lim_{x \to - 8} 2 x + 7}$

Step 13

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

$\frac{2 {(lim_{x \to - 8} x)}^{2} + 7}{lim_{x \to - 8} x - 1 \cdot 5} + \frac{3 - 4 {(lim_{x \to - 8} x)}^{2}}{lim_{x \to - 8} 2 x + 7}$

Step 14

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

$\frac{2 {(lim_{x \to - 8} x)}^{2} + 7}{lim_{x \to - 8} x - 1 \cdot 5} + \frac{3 - 4 {(lim_{x \to - 8} x)}^{2}}{lim_{x \to - 8} 2 x + lim_{x \to - 8} 7}$

Step 15

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

$\frac{2 {(lim_{x \to - 8} x)}^{2} + 7}{lim_{x \to - 8} x - 1 \cdot 5} + \frac{3 - 4 {(lim_{x \to - 8} x)}^{2}}{2 lim_{x \to - 8} x + lim_{x \to - 8} 7}$

Step 16

Evaluate the limit of $7$ which is constant as $x$ approaches $- 8$ .

$\frac{2 {(lim_{x \to - 8} x)}^{2} + 7}{lim_{x \to - 8} x - 1 \cdot 5} + \frac{3 - 4 {(lim_{x \to - 8} x)}^{2}}{2 lim_{x \to - 8} x + 7}$

Step 17

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

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Step 17.1

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

$\frac{2 {(- 8)}^{2} + 7}{lim_{x \to - 8} x - 1 \cdot 5} + \frac{3 - 4 {(lim_{x \to - 8} x)}^{2}}{2 lim_{x \to - 8} x + 7}$

Step 17.2

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

$\frac{2 {(- 8)}^{2} + 7}{- 8 - 1 \cdot 5} + \frac{3 - 4 {(lim_{x \to - 8} x)}^{2}}{2 lim_{x \to - 8} x + 7}$

Step 17.3

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

$\frac{2 {(- 8)}^{2} + 7}{- 8 - 1 \cdot 5} + \frac{3 - 4 {(- 8)}^{2}}{2 lim_{x \to - 8} x + 7}$

Step 17.4

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

$\frac{2 {(- 8)}^{2} + 7}{- 8 - 1 \cdot 5} + \frac{3 - 4 {(- 8)}^{2}}{2 \cdot - 8 + 7}$

Step 18

Simplify the answer.

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Step 18.1

Simplify each term.

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Step 18.1.1

Simplify the numerator.

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Step 18.1.1.1

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

$\frac{2 \cdot 64 + 7}{- 8 - 1 \cdot 5} + \frac{3 - 4 {(- 8)}^{2}}{2 \cdot - 8 + 7}$

Step 18.1.1.2

Multiply $2$ by $64$ .

$\frac{128 + 7}{- 8 - 1 \cdot 5} + \frac{3 - 4 {(- 8)}^{2}}{2 \cdot - 8 + 7}$

Step 18.1.1.3

Add $128$ and $7$ .

$\frac{135}{- 8 - 1 \cdot 5} + \frac{3 - 4 {(- 8)}^{2}}{2 \cdot - 8 + 7}$

Step 18.1.2

Simplify the denominator.

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Step 18.1.2.1

Multiply $- 1$ by $5$ .

$\frac{135}{- 8 - 5} + \frac{3 - 4 {(- 8)}^{2}}{2 \cdot - 8 + 7}$

Step 18.1.2.2

Subtract $5$ from $- 8$ .

$\frac{135}{- 13} + \frac{3 - 4 {(- 8)}^{2}}{2 \cdot - 8 + 7}$

Step 18.1.3

Move the negative in front of the fraction.

$- \frac{135}{13} + \frac{3 - 4 {(- 8)}^{2}}{2 \cdot - 8 + 7}$

Step 18.1.4

Simplify the numerator.

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Step 18.1.4.1

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

$- \frac{135}{13} + \frac{3 - 4 \cdot 64}{2 \cdot - 8 + 7}$

Step 18.1.4.2

Multiply $- 4$ by $64$ .

$- \frac{135}{13} + \frac{3 - 256}{2 \cdot - 8 + 7}$

Step 18.1.4.3

Subtract $256$ from $3$ .

$- \frac{135}{13} + \frac{- 253}{2 \cdot - 8 + 7}$

Step 18.1.5

Simplify the denominator.

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Step 18.1.5.1

Multiply $2$ by $- 8$ .

$- \frac{135}{13} + \frac{- 253}{- 16 + 7}$

Step 18.1.5.2

Add $- 16$ and $7$ .

$- \frac{135}{13} + \frac{- 253}{- 9}$

Step 18.1.6

Dividing two negative values results in a positive value.

$- \frac{135}{13} + \frac{253}{9}$

Step 18.2

To write $- \frac{135}{13}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$- \frac{135}{13} \cdot \frac{9}{9} + \frac{253}{9}$

Step 18.3

To write $\frac{253}{9}$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$- \frac{135}{13} \cdot \frac{9}{9} + \frac{253}{9} \cdot \frac{13}{13}$

Step 18.4

Write each expression with a common denominator of $117$ , by multiplying each by an appropriate factor of $1$ .

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Step 18.4.1

Multiply $\frac{135}{13}$ by $\frac{9}{9}$ .

$- \frac{135 \cdot 9}{13 \cdot 9} + \frac{253}{9} \cdot \frac{13}{13}$

Step 18.4.2

Multiply $13$ by $9$ .

$- \frac{135 \cdot 9}{117} + \frac{253}{9} \cdot \frac{13}{13}$

Step 18.4.3

Multiply $\frac{253}{9}$ by $\frac{13}{13}$ .

$- \frac{135 \cdot 9}{117} + \frac{253 \cdot 13}{9 \cdot 13}$

Step 18.4.4

Multiply $9$ by $13$ .

$- \frac{135 \cdot 9}{117} + \frac{253 \cdot 13}{117}$

Step 18.5

Combine the numerators over the common denominator.

$\frac{- 135 \cdot 9 + 253 \cdot 13}{117}$

Step 18.6

Simplify the numerator.

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Step 18.6.1

Multiply $- 135$ by $9$ .

$\frac{- 1215 + 253 \cdot 13}{117}$

Step 18.6.2

Multiply $253$ by $13$ .

$\frac{- 1215 + 3289}{117}$

Step 18.6.3

Add $- 1215$ and $3289$ .

$\frac{2074}{117}$

Step 19

The result can be shown in multiple forms.

Exact Form:

$\frac{2074}{117}$

Decimal Form:

$17.72649572 \dots$