Calculus Examples

$lim_{x \to - 8} \frac{9 x^{6} - x}{(x^{3} + 1) (\sqrt{9 x^{6} - x})}$

Step 1

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

$\frac{lim_{x \to - 8} 9 x^{6} - x}{lim_{x \to - 8} (x^{3} + 1) (\sqrt{9 x^{6} - x})}$

Step 2

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

$\frac{lim_{x \to - 8} 9 x^{6} - lim_{x \to - 8} x}{lim_{x \to - 8} (x^{3} + 1) (\sqrt{9 x^{6} - x})}$

Step 3

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{9 lim_{x \to - 8} x^{6} - lim_{x \to - 8} x}{lim_{x \to - 8} (x^{3} + 1) (\sqrt{9 x^{6} - x})}$

Step 4

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

$\frac{9 {(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} x}{lim_{x \to - 8} (x^{3} + 1) (\sqrt{9 x^{6} - x})}$

Step 5

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

$\frac{9 {(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} x}{lim_{x \to - 8} x^{3} + 1 \cdot lim_{x \to - 8} \sqrt{9 x^{6} - x}}$

Step 6

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

$\frac{9 {(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} x}{(lim_{x \to - 8} x^{3} + lim_{x \to - 8} 1) \cdot lim_{x \to - 8} \sqrt{9 x^{6} - x}}$

Step 7

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{9 {(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} x}{({(lim_{x \to - 8} x)}^{3} + lim_{x \to - 8} 1) \cdot lim_{x \to - 8} \sqrt{9 x^{6} - x}}$

Step 8

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

$\frac{9 {(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} x}{({(lim_{x \to - 8} x)}^{3} + 1) \cdot lim_{x \to - 8} \sqrt{9 x^{6} - x}}$

Step 9

Move the limit under the radical sign.

$\frac{9 {(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} x}{({(lim_{x \to - 8} x)}^{3} + 1) \cdot \sqrt{lim_{x \to - 8} 9 x^{6} - x}}$

Step 10

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

$\frac{9 {(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} x}{({(lim_{x \to - 8} x)}^{3} + 1) \cdot \sqrt{lim_{x \to - 8} 9 x^{6} - lim_{x \to - 8} x}}$

Step 11

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{9 {(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} x}{({(lim_{x \to - 8} x)}^{3} + 1) \cdot \sqrt{9 lim_{x \to - 8} x^{6} - lim_{x \to - 8} x}}$

Step 12

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

$\frac{9 {(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} x}{({(lim_{x \to - 8} x)}^{3} + 1) \cdot \sqrt{9 {(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} x}}$

Step 13

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

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Step 13.1

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

$\frac{9 {(- 8)}^{6} - lim_{x \to - 8} x}{({(lim_{x \to - 8} x)}^{3} + 1) \cdot \sqrt{9 {(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} x}}$

Step 13.2

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

$\frac{9 {(- 8)}^{6} + 8}{({(lim_{x \to - 8} x)}^{3} + 1) \cdot \sqrt{9 {(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} x}}$

Step 13.3

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

$\frac{9 {(- 8)}^{6} + 8}{({(- 8)}^{3} + 1) \cdot \sqrt{9 {(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} x}}$

Step 13.4

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

$\frac{9 {(- 8)}^{6} + 8}{({(- 8)}^{3} + 1) \cdot \sqrt{9 {(- 8)}^{6} - lim_{x \to - 8} x}}$

Step 13.5

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

$\frac{9 {(- 8)}^{6} + 8}{({(- 8)}^{3} + 1) \cdot \sqrt{9 {(- 8)}^{6} + 8}}$

$\frac{9 {(- 8)}^{6} + 8}{({(- 8)}^{3} + 1) \cdot \sqrt{9 {(- 8)}^{6} + 8}}$

Step 14

Simplify the answer.

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Step 14.1

Simplify the numerator.

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Step 14.1.1

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

$\frac{9 \cdot 262144 + 8}{({(- 8)}^{3} + 1) \cdot \sqrt{9 {(- 8)}^{6} + 8}}$

Step 14.1.2

Multiply $9$ by $262144$ .

$\frac{2359296 + 8}{({(- 8)}^{3} + 1) \cdot \sqrt{9 {(- 8)}^{6} + 8}}$

Step 14.1.3

Add $2359296$ and $8$ .

$\frac{2359304}{({(- 8)}^{3} + 1) \cdot \sqrt{9 {(- 8)}^{6} + 8}}$

$\frac{2359304}{({(- 8)}^{3} + 1) \cdot \sqrt{9 {(- 8)}^{6} + 8}}$

Step 14.2

Simplify the denominator.

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Step 14.2.1

Raise $- 8$ to the power of $3$ .

$\frac{2359304}{(- 512 + 1) \sqrt{9 {(- 8)}^{6} + 8}}$

Step 14.2.2

Add $- 512$ and $1$ .

$\frac{2359304}{- 511 \sqrt{9 {(- 8)}^{6} + 8}}$

Step 14.2.3

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

$\frac{2359304}{- 511 \sqrt{9 \cdot 262144 + 8}}$

Step 14.2.4

Multiply $9$ by $262144$ .

$\frac{2359304}{- 511 \sqrt{2359296 + 8}}$

Step 14.2.5

Add $2359296$ and $8$ .

$\frac{2359304}{- 511 \sqrt{2359304}}$

Step 14.2.6

Rewrite $2359304$ as $2^{2} \cdot 589826$ .

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Step 14.2.6.1

Factor $4$ out of $2359304$ .

$\frac{2359304}{- 511 \sqrt{4 (589826)}}$

Step 14.2.6.2

Rewrite $4$ as $2^{2}$ .

$\frac{2359304}{- 511 \sqrt{2^{2} \cdot 589826}}$

$\frac{2359304}{- 511 \sqrt{2^{2} \cdot 589826}}$

Step 14.2.7

Pull terms out from under the radical.

$\frac{2359304}{- 511 \cdot 2 \sqrt{589826}}$

Step 14.2.8

Multiply $- 511$ by $2$ .

$\frac{2359304}{- 1022 \sqrt{589826}}$

$\frac{2359304}{- 1022 \sqrt{589826}}$

Step 14.3

Cancel the common factor of $2359304$ and $- 1022$ .

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Step 14.3.1

Factor $2$ out of $2359304$ .

$\frac{2 (1179652)}{- 1022 \sqrt{589826}}$

Step 14.3.2

Cancel the common factors.

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Step 14.3.2.1

Factor $2$ out of $- 1022 \sqrt{589826}$ .

$\frac{2 (1179652)}{2 (- 511 \sqrt{589826})}$

Step 14.3.2.2

Cancel the common factor.

$\frac{2 \cdot 1179652}{2 (- 511 \sqrt{589826})}$

Step 14.3.2.3

Rewrite the expression.

$\frac{1179652}{- 511 \sqrt{589826}}$

$\frac{1179652}{- 511 \sqrt{589826}}$

$\frac{1179652}{- 511 \sqrt{589826}}$

Step 14.4

Move the negative in front of the fraction.

$- \frac{1179652}{511 \sqrt{589826}}$

Step 14.5

Multiply $\frac{1179652}{511 \sqrt{589826}}$ by $\frac{\sqrt{589826}}{\sqrt{589826}}$ .

$- (\frac{1179652}{511 \sqrt{589826}} \cdot \frac{\sqrt{589826}}{\sqrt{589826}})$

Step 14.6

Combine and simplify the denominator.

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Step 14.6.1

Multiply $\frac{1179652}{511 \sqrt{589826}}$ by $\frac{\sqrt{589826}}{\sqrt{589826}}$ .

$- \frac{1179652 \sqrt{589826}}{511 \sqrt{589826} \sqrt{589826}}$

Step 14.6.2

Move $\sqrt{589826}$ .

$- \frac{1179652 \sqrt{589826}}{511 (\sqrt{589826} \sqrt{589826})}$

Step 14.6.3

Raise $\sqrt{589826}$ to the power of $1$ .

$- \frac{1179652 \sqrt{589826}}{511 ({\sqrt{589826}}^{1} \sqrt{589826})}$

Step 14.6.4

Raise $\sqrt{589826}$ to the power of $1$ .

$- \frac{1179652 \sqrt{589826}}{511 ({\sqrt{589826}}^{1} {\sqrt{589826}}^{1})}$

Step 14.6.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{1179652 \sqrt{589826}}{511 {\sqrt{589826}}^{1 + 1}}$

Step 14.6.6

Add $1$ and $1$ .

$- \frac{1179652 \sqrt{589826}}{511 {\sqrt{589826}}^{2}}$

Step 14.6.7

Rewrite ${\sqrt{589826}}^{2}$ as $589826$ .

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Step 14.6.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{589826}$ as $589826^{\frac{1}{2}}$ .

$- \frac{1179652 \sqrt{589826}}{511 {(589826^{\frac{1}{2}})}^{2}}$

Step 14.6.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{1179652 \sqrt{589826}}{511 \cdot 589826^{\frac{1}{2} \cdot 2}}$

Step 14.6.7.3

Combine $\frac{1}{2}$ and $2$ .

$- \frac{1179652 \sqrt{589826}}{511 \cdot 589826^{\frac{2}{2}}}$

Step 14.6.7.4

Cancel the common factor of $2$ .

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Step 14.6.7.4.1

Cancel the common factor.

$- \frac{1179652 \sqrt{589826}}{511 \cdot 589826^{\frac{2}{2}}}$

Step 14.6.7.4.2

Rewrite the expression.

$- \frac{1179652 \sqrt{589826}}{511 \cdot 589826^{1}}$

$- \frac{1179652 \sqrt{589826}}{511 \cdot 589826^{1}}$

Step 14.6.7.5

Evaluate the exponent.

$- \frac{1179652 \sqrt{589826}}{511 \cdot 589826}$

$- \frac{1179652 \sqrt{589826}}{511 \cdot 589826}$

$- \frac{1179652 \sqrt{589826}}{511 \cdot 589826}$

Step 14.7

Cancel the common factor of $1179652$ and $589826$ .

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Step 14.7.1

Factor $589826$ out of $1179652 \sqrt{589826}$ .

$- \frac{589826 (2 \sqrt{589826})}{511 \cdot 589826}$

Step 14.7.2

Cancel the common factors.

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Step 14.7.2.1

Factor $589826$ out of $511 \cdot 589826$ .

$- \frac{589826 (2 \sqrt{589826})}{589826 \cdot 511}$

Step 14.7.2.2

Cancel the common factor.

$- \frac{589826 (2 \sqrt{589826})}{589826 \cdot 511}$

Step 14.7.2.3

Rewrite the expression.

$- \frac{2 \sqrt{589826}}{511}$

$- \frac{2 \sqrt{589826}}{511}$

$- \frac{2 \sqrt{589826}}{511}$

$- \frac{2 \sqrt{589826}}{511}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$- \frac{2 \sqrt{589826}}{511}$

Decimal Form:

$- 3.00587593 \dots$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$