Calculus Examples

$\frac{lim_{x \to 8} 3 \sqrt{x^{3}} + 3 x}{\sqrt{2 x^{3}}}$

Step 1

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

$\frac{lim_{x \to 8} 3 \sqrt{x^{3}} + lim_{x \to 8} 3 x}{\sqrt{2 x^{3}}}$

Step 2

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

$\frac{3 lim_{x \to 8} \sqrt{x^{3}} + lim_{x \to 8} 3 x}{\sqrt{2 x^{3}}}$

Step 3

Move the limit under the radical sign.

$\frac{3 \sqrt{lim_{x \to 8} x^{3}} + lim_{x \to 8} 3 x}{\sqrt{2 x^{3}}}$

Step 4

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

$\frac{3 \sqrt{{(lim_{x \to 8} x)}^{3}} + lim_{x \to 8} 3 x}{\sqrt{2 x^{3}}}$

Step 5

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

$\frac{3 \sqrt{{(lim_{x \to 8} x)}^{3}} + 3 lim_{x \to 8} x}{\sqrt{2 x^{3}}}$

Step 6

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

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Step 6.1

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

$\frac{3 \sqrt{8^{3}} + 3 lim_{x \to 8} x}{\sqrt{2 x^{3}}}$

Step 6.2

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

$\frac{3 \sqrt{8^{3}} + 3 \cdot 8}{\sqrt{2 x^{3}}}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$\frac{3 \sqrt{512} + 3 \cdot 8}{\sqrt{2 x^{3}}}$

Step 7.1.2

Rewrite $512$ as $16^{2} \cdot 2$ .

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Step 7.1.2.1

Factor $256$ out of $512$ .

$\frac{3 \sqrt{256 (2)} + 3 \cdot 8}{\sqrt{2 x^{3}}}$

Step 7.1.2.2

Rewrite $256$ as $16^{2}$ .

$\frac{3 \sqrt{16^{2} \cdot 2} + 3 \cdot 8}{\sqrt{2 x^{3}}}$

Step 7.1.3

Pull terms out from under the radical.

$\frac{3 (16 \sqrt{2}) + 3 \cdot 8}{\sqrt{2 x^{3}}}$

Step 7.1.4

Multiply $16$ by $3$ .

$\frac{48 \sqrt{2} + 3 \cdot 8}{\sqrt{2 x^{3}}}$

Step 7.1.5

Multiply $3$ by $8$ .

$\frac{48 \sqrt{2} + 24}{\sqrt{2 x^{3}}}$

Step 7.2

Simplify the denominator.

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Step 7.2.1

Rewrite $2 x^{3}$ as $x^{2} \cdot (2 x)$ .

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Step 7.2.1.1

Factor out $x^{2}$ .

$\frac{48 \sqrt{2} + 24}{\sqrt{2 (x^{2} x)}}$

Step 7.2.1.2

Reorder $2$ and $x^{2}$ .

$\frac{48 \sqrt{2} + 24}{\sqrt{x^{2} \cdot 2 x}}$

Step 7.2.1.3

Add parentheses.

$\frac{48 \sqrt{2} + 24}{\sqrt{x^{2} \cdot (2 x)}}$

Step 7.2.2

Pull terms out from under the radical.

$\frac{48 \sqrt{2} + 24}{x \sqrt{2 x}}$

Step 7.3

Multiply $\frac{48 \sqrt{2} + 24}{x \sqrt{2 x}}$ by $\frac{\sqrt{2 x}}{\sqrt{2 x}}$ .

$\frac{48 \sqrt{2} + 24}{x \sqrt{2 x}} \cdot \frac{\sqrt{2 x}}{\sqrt{2 x}}$

Step 7.4

Combine and simplify the denominator.

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Step 7.4.1

Multiply $\frac{48 \sqrt{2} + 24}{x \sqrt{2 x}}$ by $\frac{\sqrt{2 x}}{\sqrt{2 x}}$ .

$\frac{(48 \sqrt{2} + 24) \sqrt{2 x}}{x \sqrt{2 x} \sqrt{2 x}}$

Step 7.4.2

Move $\sqrt{2 x}$ .

$\frac{(48 \sqrt{2} + 24) \sqrt{2 x}}{x (\sqrt{2 x} \sqrt{2 x})}$

Step 7.4.3

Raise $\sqrt{2 x}$ to the power of $1$ .

$\frac{(48 \sqrt{2} + 24) \sqrt{2 x}}{x ({\sqrt{2 x}}^{1} \sqrt{2 x})}$

Step 7.4.4

Raise $\sqrt{2 x}$ to the power of $1$ .

$\frac{(48 \sqrt{2} + 24) \sqrt{2 x}}{x ({\sqrt{2 x}}^{1} {\sqrt{2 x}}^{1})}$

Step 7.4.5

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

$\frac{(48 \sqrt{2} + 24) \sqrt{2 x}}{x {\sqrt{2 x}}^{1 + 1}}$

Step 7.4.6

Add $1$ and $1$ .

$\frac{(48 \sqrt{2} + 24) \sqrt{2 x}}{x {\sqrt{2 x}}^{2}}$

Step 7.4.7

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

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Step 7.4.7.1

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

$\frac{(48 \sqrt{2} + 24) \sqrt{2 x}}{x {({(2 x)}^{\frac{1}{2}})}^{2}}$

Step 7.4.7.2

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

$\frac{(48 \sqrt{2} + 24) \sqrt{2 x}}{x {(2 x)}^{\frac{1}{2} \cdot 2}}$

Step 7.4.7.3

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

$\frac{(48 \sqrt{2} + 24) \sqrt{2 x}}{x {(2 x)}^{\frac{2}{2}}}$

Step 7.4.7.4

Cancel the common factor of $2$ .

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Step 7.4.7.4.1

Cancel the common factor.

$\frac{(48 \sqrt{2} + 24) \sqrt{2 x}}{x {(2 x)}^{\frac{2}{2}}}$

Step 7.4.7.4.2

Rewrite the expression.

$\frac{(48 \sqrt{2} + 24) \sqrt{2 x}}{x {(2 x)}^{1}}$

Step 7.4.7.5

Simplify.

$\frac{(48 \sqrt{2} + 24) \sqrt{2 x}}{x (2 x)}$

Step 7.5

Multiply $x$ by $x$ by adding the exponents.

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Step 7.5.1

Move $x$ .

$\frac{(48 \sqrt{2} + 24) \sqrt{2 x}}{x \cdot x \cdot 2}$

Step 7.5.2

Multiply $x$ by $x$ .

$\frac{(48 \sqrt{2} + 24) \sqrt{2 x}}{x^{2} \cdot 2}$

Step 7.6

Cancel the common factor of $48 \sqrt{2} + 24$ and $2$ .

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Step 7.6.1

Factor $2$ out of $(48 \sqrt{2} + 24) \sqrt{2 x}$ .

$\frac{2 ((24 \sqrt{2} + 12) \sqrt{2 x})}{x^{2} \cdot 2}$

Step 7.6.2

Cancel the common factors.

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Step 7.6.2.1

Factor $2$ out of $x^{2} \cdot 2$ .

$\frac{2 ((24 \sqrt{2} + 12) \sqrt{2 x})}{2 \cdot x^{2}}$

Step 7.6.2.2

Cancel the common factor.

$\frac{2 ((24 \sqrt{2} + 12) \sqrt{2 x})}{2 \cdot x^{2}}$

Step 7.6.2.3

Rewrite the expression.

$\frac{(24 \sqrt{2} + 12) \sqrt{2 x}}{x^{2}}$

Step 7.7

Apply the distributive property.

$\frac{24 \sqrt{2} \sqrt{2 x} + 12 \sqrt{2 x}}{x^{2}}$

Step 7.8

Multiply $24 \sqrt{2} \sqrt{2 x}$ .

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Step 7.8.1

Combine using the product rule for radicals.

$\frac{24 \sqrt{2 x \cdot 2} + 12 \sqrt{2 x}}{x^{2}}$

Step 7.8.2

Multiply $2$ by $2$ .

$\frac{24 \sqrt{4 x} + 12 \sqrt{2 x}}{x^{2}}$

Step 7.9

Simplify each term.

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Step 7.9.1

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

$\frac{24 \sqrt{2^{2} x} + 12 \sqrt{2 x}}{x^{2}}$

Step 7.9.2

Pull terms out from under the radical.

$\frac{24 (2 \sqrt{x}) + 12 \sqrt{2 x}}{x^{2}}$

Step 7.9.3

Multiply $2$ by $24$ .

$\frac{48 \sqrt{x} + 12 \sqrt{2 x}}{x^{2}}$

Step 7.10

Simplify the numerator.

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Step 7.10.1

Factor $12$ out of $48 \sqrt{x} + 12 \sqrt{2 x}$ .

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Step 7.10.1.1

Factor $12$ out of $48 \sqrt{x}$ .

$\frac{12 (4 \sqrt{x}) + 12 \sqrt{2 x}}{x^{2}}$

Step 7.10.1.2

Factor $12$ out of $12 \sqrt{2 x}$ .

$\frac{12 (4 \sqrt{x}) + 12 (\sqrt{2 x})}{x^{2}}$

Step 7.10.1.3

Factor $12$ out of $12 (4 \sqrt{x}) + 12 (\sqrt{2 x})$ .

$\frac{12 (4 \sqrt{x} + \sqrt{2 x})}{x^{2}}$

Step 7.10.2

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

$\frac{12 (4 x^{\frac{1}{2}} + \sqrt{2 x})}{x^{2}}$

Step 7.10.3

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

$\frac{12 (4 x^{\frac{1}{2}} + {(2 x)}^{\frac{1}{2}})}{x^{2}}$

Step 7.10.4

Apply the product rule to $2 x$ .

$\frac{12 (4 x^{\frac{1}{2}} + 2^{\frac{1}{2}} x^{\frac{1}{2}})}{x^{2}}$

Step 7.10.5

Factor $x^{\frac{1}{2}}$ out of $4 x^{\frac{1}{2}} + 2^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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Step 7.10.5.1

Factor $x^{\frac{1}{2}}$ out of $4 x^{\frac{1}{2}}$ .

$\frac{12 (x^{\frac{1}{2}} \cdot 4 + 2^{\frac{1}{2}} x^{\frac{1}{2}})}{x^{2}}$

Step 7.10.5.2

Factor $x^{\frac{1}{2}}$ out of $2^{\frac{1}{2}} x^{\frac{1}{2}}$ .

$\frac{12 (x^{\frac{1}{2}} \cdot 4 + x^{\frac{1}{2}} \cdot 2^{\frac{1}{2}})}{x^{2}}$

Step 7.10.5.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} \cdot 4 + x^{\frac{1}{2}} \cdot 2^{\frac{1}{2}}$ .

$\frac{12 (x^{\frac{1}{2}} (4 + 2^{\frac{1}{2}}))}{x^{2}}$

$\frac{12 x^{\frac{1}{2}} (4 + 2^{\frac{1}{2}})}{x^{2}}$

Step 7.11

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{12 (4 + 2^{\frac{1}{2}})}{x^{2} x^{- \frac{1}{2}}}$

Step 7.12

Multiply $x^{2}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 7.12.1

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

$\frac{12 (4 + 2^{\frac{1}{2}})}{x^{2 - \frac{1}{2}}}$

Step 7.12.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{12 (4 + 2^{\frac{1}{2}})}{x^{2 \cdot \frac{2}{2} - \frac{1}{2}}}$

Step 7.12.3

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

$\frac{12 (4 + 2^{\frac{1}{2}})}{x^{\frac{2 \cdot 2}{2} - \frac{1}{2}}}$

Step 7.12.4

Combine the numerators over the common denominator.

$\frac{12 (4 + 2^{\frac{1}{2}})}{x^{\frac{2 \cdot 2 - 1}{2}}}$

Step 7.12.5

Simplify the numerator.

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Step 7.12.5.1

Multiply $2$ by $2$ .

$\frac{12 (4 + 2^{\frac{1}{2}})}{x^{\frac{4 - 1}{2}}}$

Step 7.12.5.2

Subtract $1$ from $4$ .

$\frac{12 (4 + 2^{\frac{1}{2}})}{x^{\frac{3}{2}}}$