Calculus Examples

$lim_{x \to 8} \frac{4 x^{3}}{2 x^{3} \sqrt{9 x^{6} + 15 x^{4}}}$

Step 1

Reduce.

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Step 1.1

Cancel the common factor of $4$ and $2$ .

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Step 1.1.1

Factor $2$ out of $4 x^{3}$ .

$lim_{x \to 8} \frac{2 (2 x^{3})}{2 x^{3} \sqrt{9 x^{6} + 15 x^{4}}}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $2$ out of $2 x^{3} \sqrt{9 x^{6} + 15 x^{4}}$ .

$lim_{x \to 8} \frac{2 (2 x^{3})}{2 (x^{3} \sqrt{9 x^{6} + 15 x^{4}})}$

Step 1.1.2.2

Cancel the common factor.

$lim_{x \to 8} \frac{2 (2 x^{3})}{2 (x^{3} \sqrt{9 x^{6} + 15 x^{4}})}$

Step 1.1.2.3

Rewrite the expression.

$lim_{x \to 8} \frac{2 x^{3}}{x^{3} \sqrt{9 x^{6} + 15 x^{4}}}$

Step 1.2

Cancel the common factor of $x^{3}$ .

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Step 1.2.1

Cancel the common factor.

$lim_{x \to 8} \frac{2 x^{3}}{x^{3} \sqrt{9 x^{6} + 15 x^{4}}}$

Step 1.2.2

Rewrite the expression.

$lim_{x \to 8} \frac{2}{\sqrt{9 x^{6} + 15 x^{4}}}$

Step 2

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

$2 lim_{x \to 8} \frac{1}{\sqrt{9 x^{6} + 15 x^{4}}}$

Step 3

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

$2 \frac{lim_{x \to 8} 1}{lim_{x \to 8} \sqrt{9 x^{6} + 15 x^{4}}}$

Step 4

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

$2 \frac{1}{lim_{x \to 8} \sqrt{9 x^{6} + 15 x^{4}}}$

Step 5

Move the limit under the radical sign.

$2 \frac{1}{\sqrt{lim_{x \to 8} 9 x^{6} + 15 x^{4}}}$

Step 6

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

$2 \frac{1}{\sqrt{lim_{x \to 8} 9 x^{6} + lim_{x \to 8} 15 x^{4}}}$

Step 7

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

$2 \frac{1}{\sqrt{9 lim_{x \to 8} x^{6} + lim_{x \to 8} 15 x^{4}}}$

Step 8

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

$2 \frac{1}{\sqrt{9 {(lim_{x \to 8} x)}^{6} + lim_{x \to 8} 15 x^{4}}}$

Step 9

Move the term $15$ outside of the limit because it is constant with respect to $x$ .

$2 \frac{1}{\sqrt{9 {(lim_{x \to 8} x)}^{6} + 15 lim_{x \to 8} x^{4}}}$

Step 10

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

$2 \frac{1}{\sqrt{9 {(lim_{x \to 8} x)}^{6} + 15 {(lim_{x \to 8} x)}^{4}}}$

Step 11

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

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Step 11.1

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

$2 \frac{1}{\sqrt{9 \cdot 8^{6} + 15 {(lim_{x \to 8} x)}^{4}}}$

Step 11.2

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

$2 \frac{1}{\sqrt{9 \cdot 8^{6} + 15 \cdot 8^{4}}}$

Step 12

Simplify the answer.

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Step 12.1

Simplify the denominator.

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Step 12.1.1

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

$2 \frac{1}{\sqrt{9 \cdot 262144 + 15 \cdot 8^{4}}}$

Step 12.1.2

Multiply $9$ by $262144$ .

$2 \frac{1}{\sqrt{2359296 + 15 \cdot 8^{4}}}$

Step 12.1.3

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

$2 \frac{1}{\sqrt{2359296 + 15 \cdot 4096}}$

Step 12.1.4

Multiply $15$ by $4096$ .

$2 \frac{1}{\sqrt{2359296 + 61440}}$

Step 12.1.5

Add $2359296$ and $61440$ .

$2 \frac{1}{\sqrt{2420736}}$

Step 12.1.6

Rewrite $2420736$ as $64^{2} \cdot 591$ .

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Step 12.1.6.1

Factor $4096$ out of $2420736$ .

$2 \frac{1}{\sqrt{4096 (591)}}$

Step 12.1.6.2

Rewrite $4096$ as $64^{2}$ .

$2 \frac{1}{\sqrt{64^{2} \cdot 591}}$

Step 12.1.7

Pull terms out from under the radical.

$2 \frac{1}{64 \sqrt{591}}$

Step 12.2

Cancel the common factor of $2$ .

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Step 12.2.1

Factor $2$ out of $64 \sqrt{591}$ .

$2 \frac{1}{2 (32 \sqrt{591})}$

Step 12.2.2

Cancel the common factor.

$2 \frac{1}{2 (32 \sqrt{591})}$

Step 12.2.3

Rewrite the expression.

$\frac{1}{32 \sqrt{591}}$

Step 12.3

Multiply $\frac{1}{32 \sqrt{591}}$ by $\frac{\sqrt{591}}{\sqrt{591}}$ .

$\frac{1}{32 \sqrt{591}} \cdot \frac{\sqrt{591}}{\sqrt{591}}$

Step 12.4

Combine and simplify the denominator.

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Step 12.4.1

Multiply $\frac{1}{32 \sqrt{591}}$ by $\frac{\sqrt{591}}{\sqrt{591}}$ .

$\frac{\sqrt{591}}{32 \sqrt{591} \sqrt{591}}$

Step 12.4.2

Move $\sqrt{591}$ .

$\frac{\sqrt{591}}{32 (\sqrt{591} \sqrt{591})}$

Step 12.4.3

Raise $\sqrt{591}$ to the power of $1$ .

$\frac{\sqrt{591}}{32 ({\sqrt{591}}^{1} \sqrt{591})}$

Step 12.4.4

Raise $\sqrt{591}$ to the power of $1$ .

$\frac{\sqrt{591}}{32 ({\sqrt{591}}^{1} {\sqrt{591}}^{1})}$

Step 12.4.5

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

$\frac{\sqrt{591}}{32 {\sqrt{591}}^{1 + 1}}$

Step 12.4.6

Add $1$ and $1$ .

$\frac{\sqrt{591}}{32 {\sqrt{591}}^{2}}$

Step 12.4.7

Rewrite ${\sqrt{591}}^{2}$ as $591$ .

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Step 12.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{591}$ as $591^{\frac{1}{2}}$ .

$\frac{\sqrt{591}}{32 {(591^{\frac{1}{2}})}^{2}}$

Step 12.4.7.2

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

$\frac{\sqrt{591}}{32 \cdot 591^{\frac{1}{2} \cdot 2}}$

Step 12.4.7.3

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

$\frac{\sqrt{591}}{32 \cdot 591^{\frac{2}{2}}}$

Step 12.4.7.4

Cancel the common factor of $2$ .

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Step 12.4.7.4.1

Cancel the common factor.

$\frac{\sqrt{591}}{32 \cdot 591^{\frac{2}{2}}}$

Step 12.4.7.4.2

Rewrite the expression.

$\frac{\sqrt{591}}{32 \cdot 591^{1}}$

Step 12.4.7.5

Evaluate the exponent.

$\frac{\sqrt{591}}{32 \cdot 591}$

Step 12.5

Multiply $32$ by $591$ .

$\frac{\sqrt{591}}{18912}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{591}}{18912}$

Decimal Form:

$0.00128545 \dots$