Calculus Examples

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

Step 1

Move the exponent $7$ from ${(\frac{1 - x^{4}}{x^{2} + 5 x})}^{7}$ outside the limit using the Limits Power Rule.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $- 8$ to the power of $4$ .

${(\frac{1 - 1 \cdot 4096}{{(- 8)}^{2} + 5 \cdot - 8})}^{7}$

Step 10.1.2

Multiply $- 1$ by $4096$ .

${(\frac{1 - 4096}{{(- 8)}^{2} + 5 \cdot - 8})}^{7}$

Step 10.1.3

Subtract $4096$ from $1$ .

${(\frac{- 4095}{{(- 8)}^{2} + 5 \cdot - 8})}^{7}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

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

${(\frac{- 4095}{64 + 5 \cdot - 8})}^{7}$

Step 10.2.2

Multiply $5$ by $- 8$ .

${(\frac{- 4095}{64 - 40})}^{7}$

Step 10.2.3

Subtract $40$ from $64$ .

${(\frac{- 4095}{24})}^{7}$

Step 10.3

Cancel the common factor of $- 4095$ and $24$ .

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Step 10.3.1

Factor $3$ out of $- 4095$ .

${(\frac{3 (- 1365)}{24})}^{7}$

Step 10.3.2

Cancel the common factors.

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Step 10.3.2.1

Factor $3$ out of $24$ .

${(\frac{3 \cdot - 1365}{3 \cdot 8})}^{7}$

Step 10.3.2.2

Cancel the common factor.

${(\frac{3 \cdot - 1365}{3 \cdot 8})}^{7}$

Step 10.3.2.3

Rewrite the expression.

${(\frac{- 1365}{8})}^{7}$

Step 10.4

Move the negative in front of the fraction.

${(- \frac{1365}{8})}^{7}$

Step 10.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 10.5.1

Apply the product rule to $- \frac{1365}{8}$ .

${(- 1)}^{7} {(\frac{1365}{8})}^{7}$

Step 10.5.2

Apply the product rule to $\frac{1365}{8}$ .

${(- 1)}^{7} \frac{1365^{7}}{8^{7}}$

Step 10.6

Raise $- 1$ to the power of $7$ .

$- \frac{1365^{7}}{8^{7}}$

Step 10.7

Raise $1365$ to the power of $7$ .

$- \frac{8829346479175290000000}{8^{7}}$

Step 10.8

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

$- \frac{8829346479175290000000}{2097152}$

Step 10.9

Cancel the common factor of $8829346479175290000000$ and $2097152$ .

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Step 10.9.1

Factor $128$ out of $8829346479175290000000$ .

$- \frac{128 (68979269368556953125)}{2097152}$

Step 10.9.2

Cancel the common factors.

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Step 10.9.2.1

Factor $128$ out of $2097152$ .

$- \frac{128 \cdot 68979269368556953125}{128 \cdot 16384}$

Step 10.9.2.2

Cancel the common factor.

$- \frac{128 \cdot 68979269368556953125}{128 \cdot 16384}$

Step 10.9.2.3

Rewrite the expression.

$- \frac{68979269368556953125}{16384}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{68979269368556953125}{16384}$

Decimal Form:

$- 4.21016048 \cdot 10^{15}$