Calculus Examples

$lim_{x \to \infty} \sqrt{\frac{25 x^{2}}{4 + 36 x^{2}}}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit under the radical sign.

$\sqrt{lim_{x \to \infty} \frac{25 x^{2}}{4 + 36 x^{2}}}$

Step 1.2

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

$\sqrt{25 lim_{x \to \infty} \frac{x^{2}}{4 + 36 x^{2}}}$

Step 2

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{2}$ .

$\sqrt{25 lim_{x \to \infty} \frac{\frac{x^{2}}{x^{2}}}{\frac{4}{x^{2}} + \frac{36 x^{2}}{x^{2}}}}$

Step 3

Evaluate the limit.

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Step 3.1

Cancel the common factor of $x^{2}$ .

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Step 3.1.1

Cancel the common factor.

$\sqrt{25 lim_{x \to \infty} \frac{\frac{x^{2}}{x^{2}}}{\frac{4}{x^{2}} + \frac{36 x^{2}}{x^{2}}}}$

Step 3.1.2

Rewrite the expression.

$\sqrt{25 lim_{x \to \infty} \frac{1}{\frac{4}{x^{2}} + \frac{36 x^{2}}{x^{2}}}}$

Step 3.2

Cancel the common factor of $x^{2}$ .

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Step 3.2.1

Cancel the common factor.

$\sqrt{25 lim_{x \to \infty} \frac{1}{\frac{4}{x^{2}} + \frac{36 x^{2}}{x^{2}}}}$

Step 3.2.2

Divide $36$ by $1$ .

$\sqrt{25 lim_{x \to \infty} \frac{1}{\frac{4}{x^{2}} + 36}}$

Step 3.3

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

$\sqrt{25 \frac{lim_{x \to \infty} 1}{lim_{x \to \infty} \frac{4}{x^{2}} + 36}}$

Step 3.4

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

$\sqrt{25 \frac{1}{lim_{x \to \infty} \frac{4}{x^{2}} + 36}}$

Step 3.5

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

$\sqrt{25 \frac{1}{lim_{x \to \infty} \frac{4}{x^{2}} + lim_{x \to \infty} 36}}$

Step 3.6

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

$\sqrt{25 \frac{1}{4 lim_{x \to \infty} \frac{1}{x^{2}} + lim_{x \to \infty} 36}}$

Step 4

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\sqrt{25 \frac{1}{4 \cdot 0 + lim_{x \to \infty} 36}}$

Step 5

Evaluate the limit.

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Step 5.1

Evaluate the limit of $36$ which is constant as $x$ approaches $\infty$ .

$\sqrt{25 \frac{1}{4 \cdot 0 + 36}}$

Step 5.2

Simplify the answer.

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Step 5.2.1

Combine $25$ and $\frac{1}{4 \cdot 0 + 36}$ .

$\sqrt{\frac{25}{4 \cdot 0 + 36}}$

Step 5.2.2

Multiply $4$ by $0$ .

$\sqrt{\frac{25}{0 + 36}}$

Step 5.2.3

Add $0$ and $36$ .

$\sqrt{\frac{25}{36}}$

Step 5.2.4

Rewrite $\sqrt{\frac{25}{36}}$ as $\frac{\sqrt{25}}{\sqrt{36}}$ .

$\frac{\sqrt{25}}{\sqrt{36}}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{5^{2}}}{\sqrt{36}}$

Step 5.2.5.2

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

$\frac{5}{\sqrt{36}}$

Step 5.2.6

Simplify the denominator.

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Step 5.2.6.1

Rewrite $36$ as $6^{2}$ .

$\frac{5}{\sqrt{6^{2}}}$

Step 5.2.6.2

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

$\frac{5}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{6}$

Decimal Form:

$0.8\overline{3}$