Calculus Examples

$lim_{x \to 8} \sqrt{\frac{16 x^{2}}{7 + 49 x^{2}}}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} \frac{16 x^{2}}{7 + 49 x^{2}}}$

Step 2

Move the term $16$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{16 lim_{x \to 8} \frac{x^{2}}{7 + 49 x^{2}}}$

Step 3

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

$\sqrt{16 \frac{lim_{x \to 8} x^{2}}{lim_{x \to 8} 7 + 49 x^{2}}}$

Step 4

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

$\sqrt{16 \frac{{(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} 7 + 49 x^{2}}}$

Step 5

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

$\sqrt{16 \frac{{(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} 7 + lim_{x \to 8} 49 x^{2}}}$

Step 6

Evaluate the limit of $7$ which is constant as $x$ approaches $8$ .

$\sqrt{16 \frac{{(lim_{x \to 8} x)}^{2}}{7 + lim_{x \to 8} 49 x^{2}}}$

Step 7

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

$\sqrt{16 \frac{{(lim_{x \to 8} x)}^{2}}{7 + 49 lim_{x \to 8} x^{2}}}$

Step 8

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

$\sqrt{16 \frac{{(lim_{x \to 8} x)}^{2}}{7 + 49 {(lim_{x \to 8} x)}^{2}}}$

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 $x$ .

$\sqrt{16 \frac{8^{2}}{7 + 49 {(lim_{x \to 8} x)}^{2}}}$

Step 9.2

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

$\sqrt{16 \frac{8^{2}}{7 + 49 \cdot 8^{2}}}$

Step 10

Simplify the answer.

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Step 10.1

Combine $16$ and $\frac{8^{2}}{7 + 49 \cdot 8^{2}}$ .

$\sqrt{\frac{16 \cdot 8^{2}}{7 + 49 \cdot 8^{2}}}$

Step 10.2

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

$\sqrt{\frac{16 \cdot 8^{2}}{7 + 49 \cdot 64}}$

Step 10.3

Multiply $49$ by $64$ .

$\sqrt{\frac{16 \cdot 8^{2}}{7 + 3136}}$

Step 10.4

Add $7$ and $3136$ .

$\sqrt{\frac{16 \cdot 8^{2}}{3143}}$

Step 10.5

Simplify the numerator.

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Step 10.5.1

Rewrite $16$ as $2^{4}$ .

$\sqrt{\frac{2^{4} \cdot 8^{2}}{3143}}$

Step 10.5.2

Rewrite $8$ as $2^{3}$ .

$\sqrt{\frac{2^{4} \cdot {(2^{3})}^{2}}{3143}}$

Step 10.5.3

Multiply the exponents in ${(2^{3})}^{2}$ .

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Step 10.5.3.1

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

$\sqrt{\frac{2^{4} \cdot 2^{3 \cdot 2}}{3143}}$

Step 10.5.3.2

Multiply $3$ by $2$ .

$\sqrt{\frac{2^{4} \cdot 2^{6}}{3143}}$

Step 10.5.4

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

$\sqrt{\frac{2^{4 + 6}}{3143}}$

Step 10.5.5

Add $4$ and $6$ .

$\sqrt{\frac{2^{10}}{3143}}$

Step 10.6

Raise $2$ to the power of $10$ .

$\sqrt{\frac{1024}{3143}}$

Step 10.7

Rewrite $\sqrt{\frac{1024}{3143}}$ as $\frac{\sqrt{1024}}{\sqrt{3143}}$ .

$\frac{\sqrt{1024}}{\sqrt{3143}}$

Step 10.8

Simplify the numerator.

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Step 10.8.1

Rewrite $1024$ as $32^{2}$ .

$\frac{\sqrt{32^{2}}}{\sqrt{3143}}$

Step 10.8.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{32}{\sqrt{3143}}$

Step 10.9

Multiply $\frac{32}{\sqrt{3143}}$ by $\frac{\sqrt{3143}}{\sqrt{3143}}$ .

$\frac{32}{\sqrt{3143}} \cdot \frac{\sqrt{3143}}{\sqrt{3143}}$

Step 10.10

Combine and simplify the denominator.

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Step 10.10.1

Multiply $\frac{32}{\sqrt{3143}}$ by $\frac{\sqrt{3143}}{\sqrt{3143}}$ .

$\frac{32 \sqrt{3143}}{\sqrt{3143} \sqrt{3143}}$

Step 10.10.2

Raise $\sqrt{3143}$ to the power of $1$ .

$\frac{32 \sqrt{3143}}{{\sqrt{3143}}^{1} \sqrt{3143}}$

Step 10.10.3

Raise $\sqrt{3143}$ to the power of $1$ .

$\frac{32 \sqrt{3143}}{{\sqrt{3143}}^{1} {\sqrt{3143}}^{1}}$

Step 10.10.4

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

$\frac{32 \sqrt{3143}}{{\sqrt{3143}}^{1 + 1}}$

Step 10.10.5

Add $1$ and $1$ .

$\frac{32 \sqrt{3143}}{{\sqrt{3143}}^{2}}$

Step 10.10.6

Rewrite ${\sqrt{3143}}^{2}$ as $3143$ .

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Step 10.10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3143}$ as $3143^{\frac{1}{2}}$ .

$\frac{32 \sqrt{3143}}{{(3143^{\frac{1}{2}})}^{2}}$

Step 10.10.6.2

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

$\frac{32 \sqrt{3143}}{3143^{\frac{1}{2} \cdot 2}}$

Step 10.10.6.3

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

$\frac{32 \sqrt{3143}}{3143^{\frac{2}{2}}}$

Step 10.10.6.4

Cancel the common factor of $2$ .

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Step 10.10.6.4.1

Cancel the common factor.

$\frac{32 \sqrt{3143}}{3143^{\frac{2}{2}}}$

Step 10.10.6.4.2

Rewrite the expression.

$\frac{32 \sqrt{3143}}{3143^{1}}$

Step 10.10.6.5

Evaluate the exponent.

$\frac{32 \sqrt{3143}}{3143}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{32 \sqrt{3143}}{3143}$

Decimal Form:

$0.57079188 \dots$