Calculus Examples

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

Step 1

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

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

Step 2

Move the limit under the radical sign.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{\frac{{(lim_{x \to 8} x)}^{3}}{lim_{x \to 8} x - 1 \cdot 2}}}{lim_{x \to 8} x}$

Step 7

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

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Step 7.1

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

$\frac{\sqrt{\frac{8^{3}}{lim_{x \to 8} x - 1 \cdot 2}}}{lim_{x \to 8} x}$

Step 7.2

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

$\frac{\sqrt{\frac{8^{3}}{8 - 1 \cdot 2}}}{lim_{x \to 8} x}$

Step 7.3

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

$\frac{\sqrt{\frac{8^{3}}{8 - 1 \cdot 2}}}{8}$

Step 8

Simplify the answer.

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Step 8.1

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

$\frac{\sqrt{\frac{512}{8 - 1 \cdot 2}}}{8}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $- 1$ by $2$ .

$\frac{\sqrt{\frac{512}{8 - 2}}}{8}$

Step 8.2.2

Subtract $2$ from $8$ .

$\frac{\sqrt{\frac{512}{6}}}{8}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Cancel the common factor of $512$ and $6$ .

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Step 8.3.1.1

Factor $2$ out of $512$ .

$\frac{\sqrt{\frac{2 (256)}{6}}}{8}$

Step 8.3.1.2

Cancel the common factors.

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Step 8.3.1.2.1

Factor $2$ out of $6$ .

$\frac{\sqrt{\frac{2 \cdot 256}{2 \cdot 3}}}{8}$

Step 8.3.1.2.2

Cancel the common factor.

$\frac{\sqrt{\frac{2 \cdot 256}{2 \cdot 3}}}{8}$

Step 8.3.1.2.3

Rewrite the expression.

$\frac{\sqrt{\frac{256}{3}}}{8}$

Step 8.3.2

Rewrite $\sqrt{\frac{256}{3}}$ as $\frac{\sqrt{256}}{\sqrt{3}}$ .

$\frac{\frac{\sqrt{256}}{\sqrt{3}}}{8}$

Step 8.3.3

Simplify the numerator.

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Step 8.3.3.1

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

$\frac{\frac{\sqrt{16^{2}}}{\sqrt{3}}}{8}$

Step 8.3.3.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\frac{16}{\sqrt{3}}}{8}$

Step 8.3.4

Multiply $\frac{16}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\frac{16}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}}{8}$

Step 8.3.5

Combine and simplify the denominator.

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Step 8.3.5.1

Multiply $\frac{16}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\frac{16 \sqrt{3}}{\sqrt{3} \sqrt{3}}}{8}$

Step 8.3.5.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\frac{16 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}}{8}$

Step 8.3.5.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\frac{16 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}}{8}$

Step 8.3.5.4

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

$\frac{\frac{16 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}}{8}$

Step 8.3.5.5

Add $1$ and $1$ .

$\frac{\frac{16 \sqrt{3}}{{\sqrt{3}}^{2}}}{8}$

Step 8.3.5.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 8.3.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\frac{16 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}}{8}$

Step 8.3.5.6.2

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

$\frac{\frac{16 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}}{8}$

Step 8.3.5.6.3

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

$\frac{\frac{16 \sqrt{3}}{3^{\frac{2}{2}}}}{8}$

Step 8.3.5.6.4

Cancel the common factor of $2$ .

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Step 8.3.5.6.4.1

Cancel the common factor.

$\frac{\frac{16 \sqrt{3}}{3^{\frac{2}{2}}}}{8}$

Step 8.3.5.6.4.2

Rewrite the expression.

$\frac{\frac{16 \sqrt{3}}{3^{1}}}{8}$

Step 8.3.5.6.5

Evaluate the exponent.

$\frac{\frac{16 \sqrt{3}}{3}}{8}$

Step 8.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{16 \sqrt{3}}{3} \cdot \frac{1}{8}$

Step 8.5

Cancel the common factor of $8$ .

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Step 8.5.1

Factor $8$ out of $16 \sqrt{3}$ .

$\frac{8 (2 \sqrt{3})}{3} \cdot \frac{1}{8}$

Step 8.5.2

Cancel the common factor.

$\frac{8 (2 \sqrt{3})}{3} \cdot \frac{1}{8}$

Step 8.5.3

Rewrite the expression.

$\frac{2 \sqrt{3}}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{3}}{3}$

Decimal Form:

$1.15470053 \dots$