Precalculus Examples

$lim_{x \to 8} \sqrt{\frac{3 x + 2}{6 x - 8}}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} \frac{3 x + 2}{6 x - 8}}$

Step 2

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

$\sqrt{\frac{lim_{x \to 8} 3 x + 2}{lim_{x \to 8} 6 x - 8}}$

Step 3

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

$\sqrt{\frac{lim_{x \to 8} 3 x + lim_{x \to 8} 2}{lim_{x \to 8} 6 x - 8}}$

Step 4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{\frac{3 lim_{x \to 8} x + lim_{x \to 8} 2}{lim_{x \to 8} 6 x - 8}}$

Step 5

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

$\sqrt{\frac{3 lim_{x \to 8} x + 2}{lim_{x \to 8} 6 x - 8}}$

Step 6

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

$\sqrt{\frac{3 lim_{x \to 8} x + 2}{lim_{x \to 8} 6 x - lim_{x \to 8} 8}}$

Step 7

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{\frac{3 lim_{x \to 8} x + 2}{6 lim_{x \to 8} x - lim_{x \to 8} 8}}$

Step 8

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

$\sqrt{\frac{3 lim_{x \to 8} x + 2}{6 lim_{x \to 8} x - 1 \cdot 8}}$

Step 9

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

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Step 9.1

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

$\sqrt{\frac{3 \cdot 8 + 2}{6 lim_{x \to 8} x - 1 \cdot 8}}$

Step 9.2

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

$\sqrt{\frac{3 \cdot 8 + 2}{6 \cdot 8 - 1 \cdot 8}}$

Step 10

Simplify the answer.

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Step 10.1

Multiply $3$ by $8$ .

$\sqrt{\frac{24 + 2}{6 \cdot 8 - 1 \cdot 8}}$

Step 10.2

Add $24$ and $2$ .

$\sqrt{\frac{26}{6 \cdot 8 - 1 \cdot 8}}$

Step 10.3

Multiply $6$ by $8$ .

$\sqrt{\frac{26}{48 - 1 \cdot 8}}$

Step 10.4

Multiply $- 1$ by $8$ .

$\sqrt{\frac{26}{48 - 8}}$

Step 10.5

Subtract $8$ from $48$ .

$\sqrt{\frac{26}{40}}$

Step 10.6

Cancel the common factor of $26$ and $40$ .

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Step 10.6.1

Factor $2$ out of $26$ .

$\sqrt{\frac{2 (13)}{40}}$

Step 10.6.2

Cancel the common factors.

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Step 10.6.2.1

Factor $2$ out of $40$ .

$\sqrt{\frac{2 \cdot 13}{2 \cdot 20}}$

Step 10.6.2.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 13}{2 \cdot 20}}$

Step 10.6.2.3

Rewrite the expression.

$\sqrt{\frac{13}{20}}$

Step 10.7

Rewrite $\sqrt{\frac{13}{20}}$ as $\frac{\sqrt{13}}{\sqrt{20}}$ .

$\frac{\sqrt{13}}{\sqrt{20}}$

Step 10.8

Simplify the denominator.

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Step 10.8.1

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 10.8.1.1

Factor $4$ out of $20$ .

$\frac{\sqrt{13}}{\sqrt{4 (5)}}$

Step 10.8.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{13}}{\sqrt{2^{2} \cdot 5}}$

Step 10.8.2

Pull terms out from under the radical.

$\frac{\sqrt{13}}{2 \sqrt{5}}$

Step 10.9

Multiply $\frac{\sqrt{13}}{2 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{13}}{2 \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 10.10

Combine and simplify the denominator.

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Step 10.10.1

Multiply $\frac{\sqrt{13}}{2 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{13} \sqrt{5}}{2 \sqrt{5} \sqrt{5}}$

Step 10.10.2

Move $\sqrt{5}$ .

$\frac{\sqrt{13} \sqrt{5}}{2 (\sqrt{5} \sqrt{5})}$

Step 10.10.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{13} \sqrt{5}}{2 ({\sqrt{5}}^{1} \sqrt{5})}$

Step 10.10.4

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{13} \sqrt{5}}{2 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}$

Step 10.10.5

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

$\frac{\sqrt{13} \sqrt{5}}{2 {\sqrt{5}}^{1 + 1}}$

Step 10.10.6

Add $1$ and $1$ .

$\frac{\sqrt{13} \sqrt{5}}{2 {\sqrt{5}}^{2}}$

Step 10.10.7

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 10.10.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt{13} \sqrt{5}}{2 {(5^{\frac{1}{2}})}^{2}}$

Step 10.10.7.2

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

$\frac{\sqrt{13} \sqrt{5}}{2 \cdot 5^{\frac{1}{2} \cdot 2}}$

Step 10.10.7.3

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

$\frac{\sqrt{13} \sqrt{5}}{2 \cdot 5^{\frac{2}{2}}}$

Step 10.10.7.4

Cancel the common factor of $2$ .

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Step 10.10.7.4.1

Cancel the common factor.

$\frac{\sqrt{13} \sqrt{5}}{2 \cdot 5^{\frac{2}{2}}}$

Step 10.10.7.4.2

Rewrite the expression.

$\frac{\sqrt{13} \sqrt{5}}{2 \cdot 5^{1}}$

Step 10.10.7.5

Evaluate the exponent.

$\frac{\sqrt{13} \sqrt{5}}{2 \cdot 5}$

Step 10.11

Simplify the numerator.

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Step 10.11.1

Combine using the product rule for radicals.

$\frac{\sqrt{13 \cdot 5}}{2 \cdot 5}$

Step 10.11.2

Multiply $13$ by $5$ .

$\frac{\sqrt{65}}{2 \cdot 5}$

Step 10.12

Multiply $2$ by $5$ .

$\frac{\sqrt{65}}{10}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{65}}{10}$

Decimal Form:

$0.80622577 \dots$