Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

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

Step 1.2

Move the limit under the radical sign.

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

Step 1.3

Evaluate the limit of $\sqrt{2}$ which is constant as $x$ approaches $2$ .

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

Step 2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

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

Step 3

Simplify the answer.

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Step 3.1

Subtract $\sqrt{2}$ from $\sqrt{2}$ .

$\frac{0}{(\sqrt{x} - \sqrt{2}) (\sqrt{x} + \sqrt{2})}$

Step 3.2

Expand $(\sqrt{x} - \sqrt{2}) (\sqrt{x} + \sqrt{2})$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$\frac{0}{\sqrt{x} (\sqrt{x} + \sqrt{2}) - \sqrt{2} (\sqrt{x} + \sqrt{2})}$

Step 3.2.2

Apply the distributive property.

$\frac{0}{\sqrt{x} \sqrt{x} + \sqrt{x} \sqrt{2} - \sqrt{2} (\sqrt{x} + \sqrt{2})}$

Step 3.2.3

Apply the distributive property.

$\frac{0}{\sqrt{x} \sqrt{x} + \sqrt{x} \sqrt{2} - \sqrt{2} \sqrt{x} - \sqrt{2} \sqrt{2}}$

Step 3.3

Combine the opposite terms in $\sqrt{x} \sqrt{x} + \sqrt{x} \sqrt{2} - \sqrt{2} \sqrt{x} - \sqrt{2} \sqrt{2}$ .

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Step 3.3.1

Reorder the factors in the terms $\sqrt{x} \sqrt{2}$ and $- \sqrt{2} \sqrt{x}$ .

$\frac{0}{\sqrt{x} \sqrt{x} + \sqrt{2} \sqrt{x} - \sqrt{2} \sqrt{x} - \sqrt{2} \sqrt{2}}$

Step 3.3.2

Subtract $\sqrt{2} \sqrt{x}$ from $\sqrt{2} \sqrt{x}$ .

$\frac{0}{\sqrt{x} \sqrt{x} + 0 - \sqrt{2} \sqrt{2}}$

Step 3.3.3

Add $\sqrt{x} \sqrt{x}$ and $0$ .

$\frac{0}{\sqrt{x} \sqrt{x} - \sqrt{2} \sqrt{2}}$

Step 3.4

Simplify each term.

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Step 3.4.1

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

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Step 3.4.1.1

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

$\frac{0}{{\sqrt{x}}^{1} \sqrt{x} - \sqrt{2} \sqrt{2}}$

Step 3.4.1.2

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

$\frac{0}{{\sqrt{x}}^{1} {\sqrt{x}}^{1} - \sqrt{2} \sqrt{2}}$

Step 3.4.1.3

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

$\frac{0}{{\sqrt{x}}^{1 + 1} - \sqrt{2} \sqrt{2}}$

Step 3.4.1.4

Add $1$ and $1$ .

$\frac{0}{{\sqrt{x}}^{2} - \sqrt{2} \sqrt{2}}$

Step 3.4.2

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

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Step 3.4.2.1

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

$\frac{0}{{(x^{\frac{1}{2}})}^{2} - \sqrt{2} \sqrt{2}}$

Step 3.4.2.2

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

$\frac{0}{x^{\frac{1}{2} \cdot 2} - \sqrt{2} \sqrt{2}}$

Step 3.4.2.3

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

$\frac{0}{x^{\frac{2}{2}} - \sqrt{2} \sqrt{2}}$

Step 3.4.2.4

Cancel the common factor of $2$ .

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Step 3.4.2.4.1

Cancel the common factor.

$\frac{0}{x^{\frac{2}{2}} - \sqrt{2} \sqrt{2}}$

Step 3.4.2.4.2

Rewrite the expression.

$\frac{0}{x^{1} - \sqrt{2} \sqrt{2}}$

Step 3.4.2.5

Simplify.

$\frac{0}{x - \sqrt{2} \sqrt{2}}$

Step 3.4.3

Multiply $- \sqrt{2} \sqrt{2}$ .

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Step 3.4.3.1

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

$\frac{0}{x - ({\sqrt{2}}^{1} \sqrt{2})}$

Step 3.4.3.2

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

$\frac{0}{x - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 3.4.3.3

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

$\frac{0}{x - {\sqrt{2}}^{1 + 1}}$

Step 3.4.3.4

Add $1$ and $1$ .

$\frac{0}{x - {\sqrt{2}}^{2}}$

Step 3.4.4

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

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Step 3.4.4.1

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

$\frac{0}{x - {(2^{\frac{1}{2}})}^{2}}$

Step 3.4.4.2

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

$\frac{0}{x - 2^{\frac{1}{2} \cdot 2}}$

Step 3.4.4.3

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

$\frac{0}{x - 2^{\frac{2}{2}}}$

Step 3.4.4.4

Cancel the common factor of $2$ .

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Step 3.4.4.4.1

Cancel the common factor.

$\frac{0}{x - 2^{\frac{2}{2}}}$

Step 3.4.4.4.2

Rewrite the expression.

$\frac{0}{x - 2^{1}}$

Step 3.4.4.5

Evaluate the exponent.

$\frac{0}{x - 1 \cdot 2}$

Step 3.4.5

Multiply $- 1$ by $2$ .

$\frac{0}{x - 2}$

Step 3.5

Divide $0$ by $x - 2$ .

$0$