Calculus Examples

$lim_{x \to 8} \sqrt{\frac{49 x^{2}}{2 + 4 x^{2}}}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} \frac{49 x^{2}}{2 + 4 x^{2}}}$

Step 2

Move the term $49$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{49 lim_{x \to 8} \frac{x^{2}}{2 + 4 x^{2}}}$

Step 3

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

$\sqrt{49 \frac{lim_{x \to 8} x^{2}}{lim_{x \to 8} 2 + 4 x^{2}}}$

Step 4

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

$\sqrt{49 \frac{{(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} 2 + 4 x^{2}}}$

Step 5

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

$\sqrt{49 \frac{{(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} 2 + lim_{x \to 8} 4 x^{2}}}$

Step 6

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

$\sqrt{49 \frac{{(lim_{x \to 8} x)}^{2}}{2 + lim_{x \to 8} 4 x^{2}}}$

Step 7

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{49 \frac{{(lim_{x \to 8} x)}^{2}}{2 + 4 lim_{x \to 8} x^{2}}}$

Step 8

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

$\sqrt{49 \frac{{(lim_{x \to 8} x)}^{2}}{2 + 4 {(lim_{x \to 8} x)}^{2}}}$

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 $x$ .

$\sqrt{49 \frac{8^{2}}{2 + 4 {(lim_{x \to 8} x)}^{2}}}$

Step 9.2

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

$\sqrt{49 \frac{8^{2}}{2 + 4 \cdot 8^{2}}}$

Step 10

Simplify the answer.

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Step 10.1

Combine $49$ and $\frac{8^{2}}{2 + 4 \cdot 8^{2}}$ .

$\sqrt{\frac{49 \cdot 8^{2}}{2 + 4 \cdot 8^{2}}}$

Step 10.2

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

$\sqrt{\frac{49 \cdot 8^{2}}{2 + 4 \cdot 64}}$

Step 10.3

Multiply $4$ by $64$ .

$\sqrt{\frac{49 \cdot 8^{2}}{2 + 256}}$

Step 10.4

Add $2$ and $256$ .

$\sqrt{\frac{49 \cdot 8^{2}}{258}}$

Step 10.5

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

$\sqrt{\frac{49 \cdot 64}{258}}$

Step 10.6

Multiply $49$ by $64$ .

$\sqrt{\frac{3136}{258}}$

Step 10.7

Cancel the common factor of $3136$ and $258$ .

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Step 10.7.1

Factor $2$ out of $3136$ .

$\sqrt{\frac{2 (1568)}{258}}$

Step 10.7.2

Cancel the common factors.

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Step 10.7.2.1

Factor $2$ out of $258$ .

$\sqrt{\frac{2 \cdot 1568}{2 \cdot 129}}$

Step 10.7.2.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 1568}{2 \cdot 129}}$

Step 10.7.2.3

Rewrite the expression.

$\sqrt{\frac{1568}{129}}$

Step 10.8

Rewrite $\sqrt{\frac{1568}{129}}$ as $\frac{\sqrt{1568}}{\sqrt{129}}$ .

$\frac{\sqrt{1568}}{\sqrt{129}}$

Step 10.9

Simplify the numerator.

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Step 10.9.1

Rewrite $1568$ as $28^{2} \cdot 2$ .

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Step 10.9.1.1

Factor $784$ out of $1568$ .

$\frac{\sqrt{784 (2)}}{\sqrt{129}}$

Step 10.9.1.2

Rewrite $784$ as $28^{2}$ .

$\frac{\sqrt{28^{2} \cdot 2}}{\sqrt{129}}$

Step 10.9.2

Pull terms out from under the radical.

$\frac{28 \sqrt{2}}{\sqrt{129}}$

Step 10.10

Multiply $\frac{28 \sqrt{2}}{\sqrt{129}}$ by $\frac{\sqrt{129}}{\sqrt{129}}$ .

$\frac{28 \sqrt{2}}{\sqrt{129}} \cdot \frac{\sqrt{129}}{\sqrt{129}}$

Step 10.11

Combine and simplify the denominator.

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Step 10.11.1

Multiply $\frac{28 \sqrt{2}}{\sqrt{129}}$ by $\frac{\sqrt{129}}{\sqrt{129}}$ .

$\frac{28 \sqrt{2} \sqrt{129}}{\sqrt{129} \sqrt{129}}$

Step 10.11.2

Raise $\sqrt{129}$ to the power of $1$ .

$\frac{28 \sqrt{2} \sqrt{129}}{{\sqrt{129}}^{1} \sqrt{129}}$

Step 10.11.3

Raise $\sqrt{129}$ to the power of $1$ .

$\frac{28 \sqrt{2} \sqrt{129}}{{\sqrt{129}}^{1} {\sqrt{129}}^{1}}$

Step 10.11.4

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

$\frac{28 \sqrt{2} \sqrt{129}}{{\sqrt{129}}^{1 + 1}}$

Step 10.11.5

Add $1$ and $1$ .

$\frac{28 \sqrt{2} \sqrt{129}}{{\sqrt{129}}^{2}}$

Step 10.11.6

Rewrite ${\sqrt{129}}^{2}$ as $129$ .

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Step 10.11.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{129}$ as $129^{\frac{1}{2}}$ .

$\frac{28 \sqrt{2} \sqrt{129}}{{(129^{\frac{1}{2}})}^{2}}$

Step 10.11.6.2

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

$\frac{28 \sqrt{2} \sqrt{129}}{129^{\frac{1}{2} \cdot 2}}$

Step 10.11.6.3

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

$\frac{28 \sqrt{2} \sqrt{129}}{129^{\frac{2}{2}}}$

Step 10.11.6.4

Cancel the common factor of $2$ .

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Step 10.11.6.4.1

Cancel the common factor.

$\frac{28 \sqrt{2} \sqrt{129}}{129^{\frac{2}{2}}}$

Step 10.11.6.4.2

Rewrite the expression.

$\frac{28 \sqrt{2} \sqrt{129}}{129^{1}}$

Step 10.11.6.5

Evaluate the exponent.

$\frac{28 \sqrt{2} \sqrt{129}}{129}$

Step 10.12

Simplify the numerator.

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Step 10.12.1

Combine using the product rule for radicals.

$\frac{28 \sqrt{129 \cdot 2}}{129}$

Step 10.12.2

Multiply $129$ by $2$ .

$\frac{28 \sqrt{258}}{129}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{28 \sqrt{258}}{129}$

Decimal Form:

$3.48640771 \dots$