Calculus Examples

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

Step 1

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

$\frac{1}{\sqrt{x^{2} - 2 x}} - x$

Step 2

Simplify the answer.

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Step 2.1

Simplify each term.

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Step 2.1.1

Factor $x$ out of $x^{2} - 2 x$ .

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Step 2.1.1.1

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

$\frac{1}{\sqrt{x \cdot x - 2 x}} - x$

Step 2.1.1.2

Factor $x$ out of $- 2 x$ .

$\frac{1}{\sqrt{x \cdot x + x \cdot - 2}} - x$

Step 2.1.1.3

Factor $x$ out of $x \cdot x + x \cdot - 2$ .

$\frac{1}{\sqrt{x (x - 2)}} - x$

Step 2.1.2

Multiply $\frac{1}{\sqrt{x (x - 2)}}$ by $\frac{\sqrt{x (x - 2)}}{\sqrt{x (x - 2)}}$ .

$\frac{1}{\sqrt{x (x - 2)}} \cdot \frac{\sqrt{x (x - 2)}}{\sqrt{x (x - 2)}} - x$

Step 2.1.3

Combine and simplify the denominator.

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Step 2.1.3.1

Multiply $\frac{1}{\sqrt{x (x - 2)}}$ by $\frac{\sqrt{x (x - 2)}}{\sqrt{x (x - 2)}}$ .

$\frac{\sqrt{x (x - 2)}}{\sqrt{x (x - 2)} \sqrt{x (x - 2)}} - x$

Step 2.1.3.2

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

$\frac{\sqrt{x (x - 2)}}{{\sqrt{x (x - 2)}}^{1} \sqrt{x (x - 2)}} - x$

Step 2.1.3.3

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

$\frac{\sqrt{x (x - 2)}}{{\sqrt{x (x - 2)}}^{1} {\sqrt{x (x - 2)}}^{1}} - x$

Step 2.1.3.4

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

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

Step 2.1.3.5

Add $1$ and $1$ .

$\frac{\sqrt{x (x - 2)}}{{\sqrt{x (x - 2)}}^{2}} - x$

Step 2.1.3.6

Rewrite ${\sqrt{x (x - 2)}}^{2}$ as $x (x - 2)$ .

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Step 2.1.3.6.1

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

$\frac{\sqrt{x (x - 2)}}{{({(x (x - 2))}^{\frac{1}{2}})}^{2}} - x$

Step 2.1.3.6.2

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

$\frac{\sqrt{x (x - 2)}}{{(x (x - 2))}^{\frac{1}{2} \cdot 2}} - x$

Step 2.1.3.6.3

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

$\frac{\sqrt{x (x - 2)}}{{(x (x - 2))}^{\frac{2}{2}}} - x$

Step 2.1.3.6.4

Cancel the common factor of $2$ .

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Step 2.1.3.6.4.1

Cancel the common factor.

$\frac{\sqrt{x (x - 2)}}{{(x (x - 2))}^{\frac{2}{2}}} - x$

Step 2.1.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{x (x - 2)}}{{(x (x - 2))}^{1}} - x$

Step 2.1.3.6.5

Simplify.

$\frac{\sqrt{x (x - 2)}}{x (x - 2)} - x$

Step 2.2

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

$\frac{\sqrt{x (x - 2)}}{x (x - 2)} - x \cdot \frac{x (x - 2)}{x (x - 2)}$

Step 2.3

Combine $- x$ and $\frac{x (x - 2)}{x (x - 2)}$ .

$\frac{\sqrt{x (x - 2)}}{x (x - 2)} + \frac{- x (x (x - 2))}{x (x - 2)}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{\sqrt{x (x - 2)} - x (x (x - 2))}{x (x - 2)}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

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Step 2.5.1.1

Move $x$ .

$\frac{\sqrt{x (x - 2)} - (x \cdot x) (x - 2)}{x (x - 2)}$

Step 2.5.1.2

Multiply $x$ by $x$ .

$\frac{\sqrt{x (x - 2)} - x^{2} (x - 2)}{x (x - 2)}$

Step 2.5.2

Apply the distributive property.

$\frac{\sqrt{x (x - 2)} - x^{2} x - x^{2} \cdot - 2}{x (x - 2)}$

Step 2.5.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.5.3.1

Move $x$ .

$\frac{\sqrt{x (x - 2)} - (x \cdot x^{2}) - x^{2} \cdot - 2}{x (x - 2)}$

Step 2.5.3.2

Multiply $x$ by $x^{2}$ .

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Step 2.5.3.2.1

Raise $x$ to the power of $1$ .

$\frac{\sqrt{x (x - 2)} - (x^{1} x^{2}) - x^{2} \cdot - 2}{x (x - 2)}$

Step 2.5.3.2.2

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

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

Step 2.5.3.3

Add $1$ and $2$ .

$\frac{\sqrt{x (x - 2)} - x^{3} - x^{2} \cdot - 2}{x (x - 2)}$

Step 2.5.4

Multiply $- 2$ by $- 1$ .

$\frac{\sqrt{x (x - 2)} - x^{3} + 2 x^{2}}{x (x - 2)}$