Calculus Examples

$lim_{x \to 4} \frac{\sqrt{2 x + 1 - 3}}{\sqrt{x - 2 \sqrt{2}}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} \sqrt{2 x + 1 - 3}}{lim_{x \to 4} \sqrt{x - 2 \sqrt{2}}}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 4} 2 x + 1 - 3}}{lim_{x \to 4} \sqrt{x - 2 \sqrt{2}}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{\sqrt{lim_{x \to 4} 2 x + lim_{x \to 4} 1 - lim_{x \to 4} 3}}{lim_{x \to 4} \sqrt{x - 2 \sqrt{2}}}$

Step 4

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

$\frac{\sqrt{2 lim_{x \to 4} x + lim_{x \to 4} 1 - lim_{x \to 4} 3}}{lim_{x \to 4} \sqrt{x - 2 \sqrt{2}}}$

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $4$ .

$\frac{\sqrt{2 lim_{x \to 4} x + 1 - lim_{x \to 4} 3}}{lim_{x \to 4} \sqrt{x - 2 \sqrt{2}}}$

Step 6

Evaluate the limit of $3$ which is constant as $x$ approaches $4$ .

$\frac{\sqrt{2 lim_{x \to 4} x + 1 - 1 \cdot 3}}{lim_{x \to 4} \sqrt{x - 2 \sqrt{2}}}$

Step 7

Move the limit under the radical sign.

$\frac{\sqrt{2 lim_{x \to 4} x + 1 - 1 \cdot 3}}{\sqrt{lim_{x \to 4} x - 2 \sqrt{2}}}$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{\sqrt{2 lim_{x \to 4} x + 1 - 1 \cdot 3}}{\sqrt{lim_{x \to 4} x - lim_{x \to 4} 2 \sqrt{2}}}$

Step 9

Evaluate the limit of $2 \sqrt{2}$ which is constant as $x$ approaches $4$ .

$\frac{\sqrt{2 lim_{x \to 4} x + 1 - 1 \cdot 3}}{\sqrt{lim_{x \to 4} x - 2 \sqrt{2}}}$

Step 10

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{\sqrt{2 \cdot 4 + 1 - 1 \cdot 3}}{\sqrt{lim_{x \to 4} x - 2 \sqrt{2}}}$

Step 10.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{\sqrt{2 \cdot 4 + 1 - 1 \cdot 3}}{\sqrt{4 - 2 \sqrt{2}}}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Multiply $2$ by $4$ .

$\frac{\sqrt{8 + 1 - 1 \cdot 3}}{\sqrt{4 - 2 \sqrt{2}}}$

Step 11.1.2

Multiply $- 1$ by $3$ .

$\frac{\sqrt{8 + 1 - 3}}{\sqrt{4 - 2 \sqrt{2}}}$

Step 11.1.3

Add $8$ and $1$ .

$\frac{\sqrt{9 - 3}}{\sqrt{4 - 2 \sqrt{2}}}$

Step 11.1.4

Subtract $3$ from $9$ .

$\frac{\sqrt{6}}{\sqrt{4 - 2 \sqrt{2}}}$

Step 11.2

Combine $\sqrt{6}$ and $\sqrt{4 - 2 \sqrt{2}}$ into a single radical.

$\sqrt{\frac{6}{4 - 2 \sqrt{2}}}$

Step 11.3

Reduce the expression $\frac{6}{4 - 2 \sqrt{2}}$ by cancelling the common factors.

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Step 11.3.1

Factor $2$ out of $6$ .

$\sqrt{\frac{2 (3)}{4 - 2 \sqrt{2}}}$

Step 11.3.2

Factor $2$ out of $4$ .

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

Step 11.3.3

Factor $2$ out of $- 2 \sqrt{2}$ .

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

Step 11.3.4

Factor $2$ out of $2 (2) + 2 (- \sqrt{2})$ .

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

Step 11.3.5

Cancel the common factor.

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

Step 11.3.6

Rewrite the expression.

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

Step 11.4

Multiply $\frac{3}{2 - \sqrt{2}}$ by $\frac{2 + \sqrt{2}}{2 + \sqrt{2}}$ .

$\sqrt{\frac{3}{2 - \sqrt{2}} \cdot \frac{2 + \sqrt{2}}{2 + \sqrt{2}}}$

Step 11.5

Multiply $\frac{3}{2 - \sqrt{2}}$ by $\frac{2 + \sqrt{2}}{2 + \sqrt{2}}$ .

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

Step 11.6

Expand the denominator using the FOIL method.

$\sqrt{\frac{3 (2 + \sqrt{2})}{4 + 2 \sqrt{2} - 2 \sqrt{2} - {\sqrt{2}}^{2}}}$

Step 11.7

Simplify.

$\sqrt{\frac{3 (2 + \sqrt{2})}{2}}$

Step 11.8

Rewrite $\sqrt{\frac{3 (2 + \sqrt{2})}{2}}$ as $\frac{\sqrt{3 (2 + \sqrt{2})}}{\sqrt{2}}$ .

$\frac{\sqrt{3 (2 + \sqrt{2})}}{\sqrt{2}}$

Step 11.9

Multiply $\frac{\sqrt{3 (2 + \sqrt{2})}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{3 (2 + \sqrt{2})}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 11.10

Combine and simplify the denominator.

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Step 11.10.1

Multiply $\frac{\sqrt{3 (2 + \sqrt{2})}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{3 (2 + \sqrt{2})} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 11.10.2

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

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

Step 11.10.3

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

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

Step 11.10.4

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

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

Step 11.10.5

Add $1$ and $1$ .

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

Step 11.10.6

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

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Step 11.10.6.1

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

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

Step 11.10.6.2

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

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

Step 11.10.6.3

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

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

Step 11.10.6.4

Cancel the common factor of $2$ .

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Step 11.10.6.4.1

Cancel the common factor.

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

Step 11.10.6.4.2

Rewrite the expression.

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

Step 11.10.6.5

Evaluate the exponent.

$\frac{\sqrt{3 (2 + \sqrt{2})} \sqrt{2}}{2}$

Step 11.11

Simplify the numerator.

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Step 11.11.1

Combine using the product rule for radicals.

$\frac{\sqrt{3 (2 + \sqrt{2}) \cdot 2}}{2}$

Step 11.11.2

Multiply $2$ by $3$ .

$\frac{\sqrt{6 (2 + \sqrt{2})}}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{6 (2 + \sqrt{2})}}{2}$

Decimal Form:

$2.26303343 \dots$