Calculus Examples

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

Step 1

Evaluate the limit of $\sqrt{1}$ which is constant as $x$ approaches $8$ .

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

Step 2

Simplify the answer.

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Step 2.1

Any root of $1$ is $1$ .

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

Step 2.2

Multiply $\frac{1}{1 - \sqrt{x}}$ by $\frac{1 + \sqrt{x}}{1 + \sqrt{x}}$ .

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

Step 2.3

Multiply $\frac{1}{1 - \sqrt{x}}$ by $\frac{1 + \sqrt{x}}{1 + \sqrt{x}}$ .

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

Step 2.4

Expand the denominator using the FOIL method.

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

Step 2.5

Simplify.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

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

Step 2.5.4.2

Rewrite the expression.

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

Step 2.5.5

Simplify.

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