Calculus Examples

$lim_{x \to 8} \frac{\sqrt{25 x^{2} - 4}}{\sqrt{9 x^{2} + 2 x + 5}}$

Step 1

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

$\frac{lim_{x \to 8} \sqrt{25 x^{2} - 4}}{lim_{x \to 8} \sqrt{9 x^{2} + 2 x + 5}}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} 25 x^{2} - 4}}{lim_{x \to 8} \sqrt{9 x^{2} + 2 x + 5}}$

Step 3

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

$\frac{\sqrt{lim_{x \to 8} 25 x^{2} - lim_{x \to 8} 4}}{lim_{x \to 8} \sqrt{9 x^{2} + 2 x + 5}}$

Step 4

Move the term $25$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{25 lim_{x \to 8} x^{2} - lim_{x \to 8} 4}}{lim_{x \to 8} \sqrt{9 x^{2} + 2 x + 5}}$

Step 5

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt{25 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 4}}{lim_{x \to 8} \sqrt{9 x^{2} + 2 x + 5}}$

Step 6

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{25 {(lim_{x \to 8} x)}^{2} - 1 \cdot 4}}{lim_{x \to 8} \sqrt{9 x^{2} + 2 x + 5}}$

Step 7

Move the limit under the radical sign.

$\frac{\sqrt{25 {(lim_{x \to 8} x)}^{2} - 1 \cdot 4}}{\sqrt{lim_{x \to 8} 9 x^{2} + 2 x + 5}}$

Step 8

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

$\frac{\sqrt{25 {(lim_{x \to 8} x)}^{2} - 1 \cdot 4}}{\sqrt{lim_{x \to 8} 9 x^{2} + lim_{x \to 8} 2 x + lim_{x \to 8} 5}}$

Step 9

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{25 {(lim_{x \to 8} x)}^{2} - 1 \cdot 4}}{\sqrt{9 lim_{x \to 8} x^{2} + lim_{x \to 8} 2 x + lim_{x \to 8} 5}}$

Step 10

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt{25 {(lim_{x \to 8} x)}^{2} - 1 \cdot 4}}{\sqrt{9 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 2 x + lim_{x \to 8} 5}}$

Step 11

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

$\frac{\sqrt{25 {(lim_{x \to 8} x)}^{2} - 1 \cdot 4}}{\sqrt{9 {(lim_{x \to 8} x)}^{2} + 2 lim_{x \to 8} x + lim_{x \to 8} 5}}$

Step 12

Evaluate the limit of $5$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{25 {(lim_{x \to 8} x)}^{2} - 1 \cdot 4}}{\sqrt{9 {(lim_{x \to 8} x)}^{2} + 2 lim_{x \to 8} x + 5}}$

Step 13

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

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Step 13.1

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

$\frac{\sqrt{25 \cdot 8^{2} - 1 \cdot 4}}{\sqrt{9 {(lim_{x \to 8} x)}^{2} + 2 lim_{x \to 8} x + 5}}$

Step 13.2

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

$\frac{\sqrt{25 \cdot 8^{2} - 1 \cdot 4}}{\sqrt{9 \cdot 8^{2} + 2 lim_{x \to 8} x + 5}}$

Step 13.3

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

$\frac{\sqrt{25 \cdot 8^{2} - 1 \cdot 4}}{\sqrt{9 \cdot 8^{2} + 2 \cdot 8 + 5}}$

Step 14

Simplify the answer.

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Step 14.1

Simplify the numerator.

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Step 14.1.1

Raise $8$ to the power of $2$ .

$\frac{\sqrt{25 \cdot 64 - 1 \cdot 4}}{\sqrt{9 \cdot 8^{2} + 2 \cdot 8 + 5}}$

Step 14.1.2

Multiply $25$ by $64$ .

$\frac{\sqrt{1600 - 1 \cdot 4}}{\sqrt{9 \cdot 8^{2} + 2 \cdot 8 + 5}}$

Step 14.1.3

Multiply $- 1$ by $4$ .

$\frac{\sqrt{1600 - 4}}{\sqrt{9 \cdot 8^{2} + 2 \cdot 8 + 5}}$

Step 14.1.4

Subtract $4$ from $1600$ .

$\frac{\sqrt{1596}}{\sqrt{9 \cdot 8^{2} + 2 \cdot 8 + 5}}$

Step 14.1.5

Rewrite $1596$ as $2^{2} \cdot 399$ .

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Step 14.1.5.1

Factor $4$ out of $1596$ .

$\frac{\sqrt{4 (399)}}{\sqrt{9 \cdot 8^{2} + 2 \cdot 8 + 5}}$

Step 14.1.5.2

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

$\frac{\sqrt{2^{2} \cdot 399}}{\sqrt{9 \cdot 8^{2} + 2 \cdot 8 + 5}}$

Step 14.1.6

Pull terms out from under the radical.

$\frac{2 \sqrt{399}}{\sqrt{9 \cdot 8^{2} + 2 \cdot 8 + 5}}$

Step 14.2

Simplify the denominator.

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Step 14.2.1

Raise $8$ to the power of $2$ .

$\frac{2 \sqrt{399}}{\sqrt{9 \cdot 64 + 2 \cdot 8 + 5}}$

Step 14.2.2

Multiply $9$ by $64$ .

$\frac{2 \sqrt{399}}{\sqrt{576 + 2 \cdot 8 + 5}}$

Step 14.2.3

Multiply $2$ by $8$ .

$\frac{2 \sqrt{399}}{\sqrt{576 + 16 + 5}}$

Step 14.2.4

Add $576$ and $16$ .

$\frac{2 \sqrt{399}}{\sqrt{592 + 5}}$

Step 14.2.5

Add $592$ and $5$ .

$\frac{2 \sqrt{399}}{\sqrt{597}}$

Step 14.3

Combine $\sqrt{399}$ and $\sqrt{597}$ into a single radical.

$2 \sqrt{\frac{399}{597}}$

Step 14.4

Cancel the common factor of $399$ and $597$ .

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Step 14.4.1

Factor $3$ out of $399$ .

$2 \sqrt{\frac{3 (133)}{597}}$

Step 14.4.2

Cancel the common factors.

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Step 14.4.2.1

Factor $3$ out of $597$ .

$2 \sqrt{\frac{3 \cdot 133}{3 \cdot 199}}$

Step 14.4.2.2

Cancel the common factor.

$2 \sqrt{\frac{3 \cdot 133}{3 \cdot 199}}$

Step 14.4.2.3

Rewrite the expression.

$2 \sqrt{\frac{133}{199}}$

Step 14.5

Rewrite $\sqrt{\frac{133}{199}}$ as $\frac{\sqrt{133}}{\sqrt{199}}$ .

$2 \frac{\sqrt{133}}{\sqrt{199}}$

Step 14.6

Multiply $\frac{\sqrt{133}}{\sqrt{199}}$ by $\frac{\sqrt{199}}{\sqrt{199}}$ .

$2 (\frac{\sqrt{133}}{\sqrt{199}} \cdot \frac{\sqrt{199}}{\sqrt{199}})$

Step 14.7

Combine and simplify the denominator.

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Step 14.7.1

Multiply $\frac{\sqrt{133}}{\sqrt{199}}$ by $\frac{\sqrt{199}}{\sqrt{199}}$ .

$2 \frac{\sqrt{133} \sqrt{199}}{\sqrt{199} \sqrt{199}}$

Step 14.7.2

Raise $\sqrt{199}$ to the power of $1$ .

$2 \frac{\sqrt{133} \sqrt{199}}{{\sqrt{199}}^{1} \sqrt{199}}$

Step 14.7.3

Raise $\sqrt{199}$ to the power of $1$ .

$2 \frac{\sqrt{133} \sqrt{199}}{{\sqrt{199}}^{1} {\sqrt{199}}^{1}}$

Step 14.7.4

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

$2 \frac{\sqrt{133} \sqrt{199}}{{\sqrt{199}}^{1 + 1}}$

Step 14.7.5

Add $1$ and $1$ .

$2 \frac{\sqrt{133} \sqrt{199}}{{\sqrt{199}}^{2}}$

Step 14.7.6

Rewrite ${\sqrt{199}}^{2}$ as $199$ .

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Step 14.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{199}$ as $199^{\frac{1}{2}}$ .

$2 \frac{\sqrt{133} \sqrt{199}}{{(199^{\frac{1}{2}})}^{2}}$

Step 14.7.6.2

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

$2 \frac{\sqrt{133} \sqrt{199}}{199^{\frac{1}{2} \cdot 2}}$

Step 14.7.6.3

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

$2 \frac{\sqrt{133} \sqrt{199}}{199^{\frac{2}{2}}}$

Step 14.7.6.4

Cancel the common factor of $2$ .

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Step 14.7.6.4.1

Cancel the common factor.

$2 \frac{\sqrt{133} \sqrt{199}}{199^{\frac{2}{2}}}$

Step 14.7.6.4.2

Rewrite the expression.

$2 \frac{\sqrt{133} \sqrt{199}}{199^{1}}$

Step 14.7.6.5

Evaluate the exponent.

$2 \frac{\sqrt{133} \sqrt{199}}{199}$

Step 14.8

Simplify the numerator.

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Step 14.8.1

Combine using the product rule for radicals.

$2 \frac{\sqrt{133 \cdot 199}}{199}$

Step 14.8.2

Multiply $133$ by $199$ .

$2 \frac{\sqrt{26467}}{199}$

Step 14.9

Combine $2$ and $\frac{\sqrt{26467}}{199}$ .

$\frac{2 \sqrt{26467}}{199}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{26467}}{199}$

Decimal Form:

$1.63504337 \dots$