Calculus Examples

$lim_{x \to - 8} \frac{\sqrt[3]{x^{6} + 64}}{3 x^{2} + \sqrt{2 x^{4} + 16}}$

Step 1

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

$\frac{lim_{x \to - 8} \sqrt[3]{x^{6} + 64}}{lim_{x \to - 8} 3 x^{2} + \sqrt{2 x^{4} + 16}}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt[3]{lim_{x \to - 8} x^{6} + 64}}{lim_{x \to - 8} 3 x^{2} + \sqrt{2 x^{4} + 16}}$

Step 3

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

$\frac{\sqrt[3]{lim_{x \to - 8} x^{6} + lim_{x \to - 8} 64}}{lim_{x \to - 8} 3 x^{2} + \sqrt{2 x^{4} + 16}}$

Step 4

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

$\frac{\sqrt[3]{{(lim_{x \to - 8} x)}^{6} + lim_{x \to - 8} 64}}{lim_{x \to - 8} 3 x^{2} + \sqrt{2 x^{4} + 16}}$

Step 5

Evaluate the limit of $64$ which is constant as $x$ approaches $- 8$ .

$\frac{\sqrt[3]{{(lim_{x \to - 8} x)}^{6} + 64}}{lim_{x \to - 8} 3 x^{2} + \sqrt{2 x^{4} + 16}}$

Step 6

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

$\frac{\sqrt[3]{{(lim_{x \to - 8} x)}^{6} + 64}}{lim_{x \to - 8} 3 x^{2} + lim_{x \to - 8} \sqrt{2 x^{4} + 16}}$

Step 7

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

$\frac{\sqrt[3]{{(lim_{x \to - 8} x)}^{6} + 64}}{3 lim_{x \to - 8} x^{2} + lim_{x \to - 8} \sqrt{2 x^{4} + 16}}$

Step 8

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

$\frac{\sqrt[3]{{(lim_{x \to - 8} x)}^{6} + 64}}{3 {(lim_{x \to - 8} x)}^{2} + lim_{x \to - 8} \sqrt{2 x^{4} + 16}}$

Step 9

Move the limit under the radical sign.

$\frac{\sqrt[3]{{(lim_{x \to - 8} x)}^{6} + 64}}{3 {(lim_{x \to - 8} x)}^{2} + \sqrt{lim_{x \to - 8} 2 x^{4} + 16}}$

Step 10

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

$\frac{\sqrt[3]{{(lim_{x \to - 8} x)}^{6} + 64}}{3 {(lim_{x \to - 8} x)}^{2} + \sqrt{lim_{x \to - 8} 2 x^{4} + lim_{x \to - 8} 16}}$

Step 11

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

$\frac{\sqrt[3]{{(lim_{x \to - 8} x)}^{6} + 64}}{3 {(lim_{x \to - 8} x)}^{2} + \sqrt{2 lim_{x \to - 8} x^{4} + lim_{x \to - 8} 16}}$

Step 12

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

$\frac{\sqrt[3]{{(lim_{x \to - 8} x)}^{6} + 64}}{3 {(lim_{x \to - 8} x)}^{2} + \sqrt{2 {(lim_{x \to - 8} x)}^{4} + lim_{x \to - 8} 16}}$

Step 13

Evaluate the limit of $16$ which is constant as $x$ approaches $- 8$ .

$\frac{\sqrt[3]{{(lim_{x \to - 8} x)}^{6} + 64}}{3 {(lim_{x \to - 8} x)}^{2} + \sqrt{2 {(lim_{x \to - 8} x)}^{4} + 16}}$

Step 14

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

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Step 14.1

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

$\frac{\sqrt[3]{{(- 8)}^{6} + 64}}{3 {(lim_{x \to - 8} x)}^{2} + \sqrt{2 {(lim_{x \to - 8} x)}^{4} + 16}}$

Step 14.2

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

$\frac{\sqrt[3]{{(- 8)}^{6} + 64}}{3 {(- 8)}^{2} + \sqrt{2 {(lim_{x \to - 8} x)}^{4} + 16}}$

Step 14.3

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

$\frac{\sqrt[3]{{(- 8)}^{6} + 64}}{3 {(- 8)}^{2} + \sqrt{2 {(- 8)}^{4} + 16}}$

Step 15

Simplify the answer.

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Step 15.1

Simplify the numerator.

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Step 15.1.1

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

$\frac{\sqrt[3]{262144 + 64}}{3 {(- 8)}^{2} + \sqrt{2 {(- 8)}^{4} + 16}}$

Step 15.1.2

Add $262144$ and $64$ .

$\frac{\sqrt[3]{262208}}{3 {(- 8)}^{2} + \sqrt{2 {(- 8)}^{4} + 16}}$

Step 15.1.3

Rewrite $262208$ as $4^{3} \cdot 4097$ .

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Step 15.1.3.1

Factor $64$ out of $262208$ .

$\frac{\sqrt[3]{64 (4097)}}{3 {(- 8)}^{2} + \sqrt{2 {(- 8)}^{4} + 16}}$

Step 15.1.3.2

Rewrite $64$ as $4^{3}$ .

$\frac{\sqrt[3]{4^{3} \cdot 4097}}{3 {(- 8)}^{2} + \sqrt{2 {(- 8)}^{4} + 16}}$

Step 15.1.4

Pull terms out from under the radical.

$\frac{4 \sqrt[3]{4097}}{3 {(- 8)}^{2} + \sqrt{2 {(- 8)}^{4} + 16}}$

Step 15.2

Simplify the denominator.

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Step 15.2.1

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

$\frac{4 \sqrt[3]{4097}}{3 \cdot 64 + \sqrt{2 {(- 8)}^{4} + 16}}$

Step 15.2.2

Multiply $3$ by $64$ .

$\frac{4 \sqrt[3]{4097}}{192 + \sqrt{2 {(- 8)}^{4} + 16}}$

Step 15.2.3

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

$\frac{4 \sqrt[3]{4097}}{192 + \sqrt{2 \cdot 4096 + 16}}$

Step 15.2.4

Multiply $2$ by $4096$ .

$\frac{4 \sqrt[3]{4097}}{192 + \sqrt{8192 + 16}}$

Step 15.2.5

Add $8192$ and $16$ .

$\frac{4 \sqrt[3]{4097}}{192 + \sqrt{8208}}$

Step 15.2.6

Rewrite $8208$ as $12^{2} \cdot 57$ .

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Step 15.2.6.1

Factor $144$ out of $8208$ .

$\frac{4 \sqrt[3]{4097}}{192 + \sqrt{144 (57)}}$

Step 15.2.6.2

Rewrite $144$ as $12^{2}$ .

$\frac{4 \sqrt[3]{4097}}{192 + \sqrt{12^{2} \cdot 57}}$

Step 15.2.7

Pull terms out from under the radical.

$\frac{4 \sqrt[3]{4097}}{192 + 12 \sqrt{57}}$

Step 15.3

Cancel the common factor of $4$ and $192 + 12 \sqrt{57}$ .

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Step 15.3.1

Factor $4$ out of $4 \sqrt[3]{4097}$ .

$\frac{4 (\sqrt[3]{4097})}{192 + 12 \sqrt{57}}$

Step 15.3.2

Cancel the common factors.

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Step 15.3.2.1

Factor $4$ out of $192$ .

$\frac{4 \sqrt[3]{4097}}{4 \cdot 48 + 12 \sqrt{57}}$

Step 15.3.2.2

Factor $4$ out of $12 \sqrt{57}$ .

$\frac{4 \sqrt[3]{4097}}{4 \cdot 48 + 4 (3 \sqrt{57})}$

Step 15.3.2.3

Factor $4$ out of $4 (48) + 4 (3 \sqrt{57})$ .

$\frac{4 \sqrt[3]{4097}}{4 (48 + 3 \sqrt{57})}$

Step 15.3.2.4

Cancel the common factor.

$\frac{4 \sqrt[3]{4097}}{4 (48 + 3 \sqrt{57})}$

Step 15.3.2.5

Rewrite the expression.

$\frac{\sqrt[3]{4097}}{48 + 3 \sqrt{57}}$

Step 15.4

Multiply $\frac{\sqrt[3]{4097}}{48 + 3 \sqrt{57}}$ by $\frac{48 - 3 \sqrt{57}}{48 - 3 \sqrt{57}}$ .

$\frac{\sqrt[3]{4097}}{48 + 3 \sqrt{57}} \cdot \frac{48 - 3 \sqrt{57}}{48 - 3 \sqrt{57}}$

Step 15.5

Multiply $\frac{\sqrt[3]{4097}}{48 + 3 \sqrt{57}}$ by $\frac{48 - 3 \sqrt{57}}{48 - 3 \sqrt{57}}$ .

$\frac{\sqrt[3]{4097} (48 - 3 \sqrt{57})}{(48 + 3 \sqrt{57}) (48 - 3 \sqrt{57})}$

Step 15.6

Expand the denominator using the FOIL method.

$\frac{\sqrt[3]{4097} (48 - 3 \sqrt{57})}{2304 - 144 \sqrt{57} + 144 \sqrt{57} - 9 {\sqrt{57}}^{2}}$

Step 15.7

Simplify.

$\frac{\sqrt[3]{4097} (48 - 3 \sqrt{57})}{1791}$

Step 15.8

Cancel the common factor of $48 - 3 \sqrt{57}$ and $1791$ .

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Step 15.8.1

Factor $3$ out of $\sqrt[3]{4097} (48 - 3 \sqrt{57})$ .

$\frac{3 (\sqrt[3]{4097} (16 - \sqrt{57}))}{1791}$

Step 15.8.2

Cancel the common factors.

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Step 15.8.2.1

Factor $3$ out of $1791$ .

$\frac{3 (\sqrt[3]{4097} (16 - \sqrt{57}))}{3 (597)}$

Step 15.8.2.2

Cancel the common factor.

$\frac{3 (\sqrt[3]{4097} (16 - \sqrt{57}))}{3 \cdot 597}$

Step 15.8.2.3

Rewrite the expression.

$\frac{\sqrt[3]{4097} (16 - \sqrt{57})}{597}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[3]{4097} (16 - \sqrt{57})}{597}$

Decimal Form:

$0.22648852 \dots$