Calculus Examples

$lim_{x \to - 8} - 5 x^{11} + x^{7} - \frac{2}{3} x^{6} - 6 x^{5} - 9$

Step 1

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

$- lim_{x \to - 8} 5 x^{11} + lim_{x \to - 8} x^{7} - lim_{x \to - 8} \frac{2}{3} x^{6} - lim_{x \to - 8} 6 x^{5} - lim_{x \to - 8} 9$

Step 2

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

$- 5 lim_{x \to - 8} x^{11} + lim_{x \to - 8} x^{7} - lim_{x \to - 8} \frac{2}{3} x^{6} - lim_{x \to - 8} 6 x^{5} - lim_{x \to - 8} 9$

Step 3

Move the exponent $11$ from $x^{11}$ outside the limit using the Limits Power Rule.

$- 5 {(lim_{x \to - 8} x)}^{11} + lim_{x \to - 8} x^{7} - lim_{x \to - 8} \frac{2}{3} x^{6} - lim_{x \to - 8} 6 x^{5} - lim_{x \to - 8} 9$

Step 4

Move the exponent $7$ from $x^{7}$ outside the limit using the Limits Power Rule.

$- 5 {(lim_{x \to - 8} x)}^{11} + {(lim_{x \to - 8} x)}^{7} - lim_{x \to - 8} \frac{2}{3} x^{6} - lim_{x \to - 8} 6 x^{5} - lim_{x \to - 8} 9$

Step 5

Move the term $\frac{2}{3}$ outside of the limit because it is constant with respect to $x$ .

$- 5 {(lim_{x \to - 8} x)}^{11} + {(lim_{x \to - 8} x)}^{7} - \frac{2}{3} lim_{x \to - 8} x^{6} - lim_{x \to - 8} 6 x^{5} - lim_{x \to - 8} 9$

Step 6

Move the exponent $6$ from $x^{6}$ outside the limit using the Limits Power Rule.

$- 5 {(lim_{x \to - 8} x)}^{11} + {(lim_{x \to - 8} x)}^{7} - \frac{2}{3} {(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} 6 x^{5} - lim_{x \to - 8} 9$

Step 7

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

$- 5 {(lim_{x \to - 8} x)}^{11} + {(lim_{x \to - 8} x)}^{7} - \frac{2}{3} {(lim_{x \to - 8} x)}^{6} - 6 lim_{x \to - 8} x^{5} - lim_{x \to - 8} 9$

Step 8

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

$- 5 {(lim_{x \to - 8} x)}^{11} + {(lim_{x \to - 8} x)}^{7} - \frac{2}{3} {(lim_{x \to - 8} x)}^{6} - 6 {(lim_{x \to - 8} x)}^{5} - lim_{x \to - 8} 9$

Step 9

Evaluate the limit of $9$ which is constant as $x$ approaches $- 8$ .

$- 5 {(lim_{x \to - 8} x)}^{11} + {(lim_{x \to - 8} x)}^{7} - \frac{2}{3} {(lim_{x \to - 8} x)}^{6} - 6 {(lim_{x \to - 8} x)}^{5} - 1 \cdot 9$

Step 10

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

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Step 10.1

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

$- 5 {(- 8)}^{11} + {(lim_{x \to - 8} x)}^{7} - \frac{2}{3} {(lim_{x \to - 8} x)}^{6} - 6 {(lim_{x \to - 8} x)}^{5} - 1 \cdot 9$

Step 10.2

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

$- 5 {(- 8)}^{11} + {(- 8)}^{7} - \frac{2}{3} {(lim_{x \to - 8} x)}^{6} - 6 {(lim_{x \to - 8} x)}^{5} - 1 \cdot 9$

Step 10.3

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

$- 5 {(- 8)}^{11} + {(- 8)}^{7} - \frac{2}{3} {(- 8)}^{6} - 6 {(lim_{x \to - 8} x)}^{5} - 1 \cdot 9$

Step 10.4

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

$- 5 {(- 8)}^{11} + {(- 8)}^{7} - \frac{2}{3} {(- 8)}^{6} - 6 {(- 8)}^{5} - 1 \cdot 9$

Step 11

Simplify the answer.

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Step 11.1

Simplify each term.

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Step 11.1.1

Raise $- 8$ to the power of $11$ .

$- 5 \cdot - 8589934592 + {(- 8)}^{7} - \frac{2}{3} {(- 8)}^{6} - 6 {(- 8)}^{5} - 1 \cdot 9$

Step 11.1.2

Multiply $- 5$ by $- 8589934592$ .

$42949672960 + {(- 8)}^{7} - \frac{2}{3} {(- 8)}^{6} - 6 {(- 8)}^{5} - 1 \cdot 9$

Step 11.1.3

Raise $- 8$ to the power of $7$ .

$42949672960 - 2097152 - \frac{2}{3} {(- 8)}^{6} - 6 {(- 8)}^{5} - 1 \cdot 9$

Step 11.1.4

Raise $- 8$ to the power of $6$ .

$42949672960 - 2097152 - \frac{2}{3} \cdot 262144 - 6 {(- 8)}^{5} - 1 \cdot 9$

Step 11.1.5

Multiply $- \frac{2}{3} \cdot 262144$ .

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Step 11.1.5.1

Multiply $262144$ by $- 1$ .

$42949672960 - 2097152 - 262144 (\frac{2}{3}) - 6 {(- 8)}^{5} - 1 \cdot 9$

Step 11.1.5.2

Combine $- 262144$ and $\frac{2}{3}$ .

$42949672960 - 2097152 + \frac{- 262144 \cdot 2}{3} - 6 {(- 8)}^{5} - 1 \cdot 9$

Step 11.1.5.3

Multiply $- 262144$ by $2$ .

$42949672960 - 2097152 + \frac{- 524288}{3} - 6 {(- 8)}^{5} - 1 \cdot 9$

Step 11.1.6

Move the negative in front of the fraction.

$42949672960 - 2097152 - \frac{524288}{3} - 6 {(- 8)}^{5} - 1 \cdot 9$

Step 11.1.7

Raise $- 8$ to the power of $5$ .

$42949672960 - 2097152 - \frac{524288}{3} - 6 \cdot - 32768 - 1 \cdot 9$

Step 11.1.8

Multiply $- 6$ by $- 32768$ .

$42949672960 - 2097152 - \frac{524288}{3} + 196608 - 1 \cdot 9$

Step 11.1.9

Multiply $- 1$ by $9$ .

$42949672960 - 2097152 - \frac{524288}{3} + 196608 - 9$

Step 11.2

Find the common denominator.

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Step 11.2.1

Write $42949672960$ as a fraction with denominator $1$ .

$\frac{42949672960}{1} - 2097152 - \frac{524288}{3} + 196608 - 9$

Step 11.2.2

Multiply $\frac{42949672960}{1}$ by $\frac{3}{3}$ .

$\frac{42949672960}{1} \cdot \frac{3}{3} - 2097152 - \frac{524288}{3} + 196608 - 9$

Step 11.2.3

Multiply $\frac{42949672960}{1}$ by $\frac{3}{3}$ .

$\frac{42949672960 \cdot 3}{3} - 2097152 - \frac{524288}{3} + 196608 - 9$

Step 11.2.4

Write $- 2097152$ as a fraction with denominator $1$ .

$\frac{42949672960 \cdot 3}{3} + \frac{- 2097152}{1} - \frac{524288}{3} + 196608 - 9$

Step 11.2.5

Multiply $\frac{- 2097152}{1}$ by $\frac{3}{3}$ .

$\frac{42949672960 \cdot 3}{3} + \frac{- 2097152}{1} \cdot \frac{3}{3} - \frac{524288}{3} + 196608 - 9$

Step 11.2.6

Multiply $\frac{- 2097152}{1}$ by $\frac{3}{3}$ .

$\frac{42949672960 \cdot 3}{3} + \frac{- 2097152 \cdot 3}{3} - \frac{524288}{3} + 196608 - 9$

Step 11.2.7

Write $196608$ as a fraction with denominator $1$ .

$\frac{42949672960 \cdot 3}{3} + \frac{- 2097152 \cdot 3}{3} - \frac{524288}{3} + \frac{196608}{1} - 9$

Step 11.2.8

Multiply $\frac{196608}{1}$ by $\frac{3}{3}$ .

$\frac{42949672960 \cdot 3}{3} + \frac{- 2097152 \cdot 3}{3} - \frac{524288}{3} + \frac{196608}{1} \cdot \frac{3}{3} - 9$

Step 11.2.9

Multiply $\frac{196608}{1}$ by $\frac{3}{3}$ .

$\frac{42949672960 \cdot 3}{3} + \frac{- 2097152 \cdot 3}{3} - \frac{524288}{3} + \frac{196608 \cdot 3}{3} - 9$

Step 11.2.10

Write $- 9$ as a fraction with denominator $1$ .

$\frac{42949672960 \cdot 3}{3} + \frac{- 2097152 \cdot 3}{3} - \frac{524288}{3} + \frac{196608 \cdot 3}{3} + \frac{- 9}{1}$

Step 11.2.11

Multiply $\frac{- 9}{1}$ by $\frac{3}{3}$ .

$\frac{42949672960 \cdot 3}{3} + \frac{- 2097152 \cdot 3}{3} - \frac{524288}{3} + \frac{196608 \cdot 3}{3} + \frac{- 9}{1} \cdot \frac{3}{3}$

Step 11.2.12

Multiply $\frac{- 9}{1}$ by $\frac{3}{3}$ .

$\frac{42949672960 \cdot 3}{3} + \frac{- 2097152 \cdot 3}{3} - \frac{524288}{3} + \frac{196608 \cdot 3}{3} + \frac{- 9 \cdot 3}{3}$

Step 11.3

Combine the numerators over the common denominator.

$\frac{42949672960 \cdot 3 - 2097152 \cdot 3 - 524288 + 196608 \cdot 3 - 9 \cdot 3}{3}$

Step 11.4

Simplify each term.

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Step 11.4.1

Multiply $42949672960$ by $3$ .

$\frac{128849018880 - 2097152 \cdot 3 - 524288 + 196608 \cdot 3 - 9 \cdot 3}{3}$

Step 11.4.2

Multiply $- 2097152$ by $3$ .

$\frac{128849018880 - 6291456 - 524288 + 196608 \cdot 3 - 9 \cdot 3}{3}$

Step 11.4.3

Multiply $196608$ by $3$ .

$\frac{128849018880 - 6291456 - 524288 + 589824 - 9 \cdot 3}{3}$

Step 11.4.4

Multiply $- 9$ by $3$ .

$\frac{128849018880 - 6291456 - 524288 + 589824 - 27}{3}$

Step 11.5

Subtract $6291456$ from $128849018880$ .

$\frac{128842727424 - 524288 + 589824 - 27}{3}$

Step 11.6

Subtract $524288$ from $128842727424$ .

$\frac{128842203136 + 589824 - 27}{3}$

Step 11.7

Add $128842203136$ and $589824$ .

$\frac{128842792960 - 27}{3}$

Step 11.8

Subtract $27$ from $128842792960$ .

$\frac{128842792933}{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{128842792933}{3}$

Decimal Form:

$42947597644.3333$