Calculus Examples

$f (x) = \frac{x}{\sqrt{25 x^{2} + 5}}$

Step 1

Find where the expression $\frac{x}{\sqrt{25 x^{2} + 5}}$ is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

Evaluate $lim_{x \to \infty} \frac{x}{\sqrt{25 x^{2} + 5}}$ to find the horizontal asymptote.

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Step 3.1

Factor $5$ out of $25 x^{2} + 5$ .

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Step 3.1.1

Factor $5$ out of $25 x^{2}$ .

$lim_{x \to \infty} \frac{x}{\sqrt{5 (5 x^{2}) + 5}}$

Step 3.1.2

Factor $5$ out of $5$ .

$lim_{x \to \infty} \frac{x}{\sqrt{5 (5 x^{2}) + 5 (1)}}$

Step 3.1.3

Factor $5$ out of $5 (5 x^{2}) + 5 (1)$ .

$lim_{x \to \infty} \frac{x}{\sqrt{5 (5 x^{2} + 1)}}$

Step 3.2

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

$lim_{x \to \infty} \frac{\frac{x}{x}}{\sqrt{\frac{5 (5 x^{2} + 1)}{x^{2}}}}$

Step 3.3

Evaluate the limit.

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Step 3.3.1

Cancel the common factor of $x$ .

$lim_{x \to \infty} \frac{1}{\sqrt{\frac{5 (5 x^{2} + 1)}{x^{2}}}}$

Step 3.3.2

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

$\frac{lim_{x \to \infty} 1}{lim_{x \to \infty} \sqrt{\frac{5 (5 x^{2} + 1)}{x^{2}}}}$

Step 3.3.3

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

$\frac{1}{lim_{x \to \infty} \sqrt{\frac{5 (5 x^{2} + 1)}{x^{2}}}}$

Step 3.3.4

Move the limit under the radical sign.

$\frac{1}{\sqrt{lim_{x \to \infty} \frac{5 (5 x^{2} + 1)}{x^{2}}}}$

Step 3.3.5

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{\sqrt{5 lim_{x \to \infty} \frac{5 x^{2} + 1}{x^{2}}}}$

Step 3.4

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

$\frac{1}{\sqrt{5 lim_{x \to \infty} \frac{\frac{5 x^{2}}{x^{2}} + \frac{1}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}$

Step 3.5

Evaluate the limit.

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Step 3.5.1

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

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Step 3.5.1.1

Cancel the common factor.

$\frac{1}{\sqrt{5 lim_{x \to \infty} \frac{\frac{5 x^{2}}{x^{2}} + \frac{1}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}$

Step 3.5.1.2

Divide $5$ by $1$ .

$\frac{1}{\sqrt{5 lim_{x \to \infty} \frac{5 + \frac{1}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}$

Step 3.5.2

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

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Step 3.5.2.1

Cancel the common factor.

$\frac{1}{\sqrt{5 lim_{x \to \infty} \frac{5 + \frac{1}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}$

Step 3.5.2.2

Rewrite the expression.

$\frac{1}{\sqrt{5 lim_{x \to \infty} \frac{5 + \frac{1}{x^{2}}}{1}}}$

Step 3.5.3

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

$\frac{1}{\sqrt{5 \frac{lim_{x \to \infty} 5 + \frac{1}{x^{2}}}{lim_{x \to \infty} 1}}}$

Step 3.5.4

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

$\frac{1}{\sqrt{5 \frac{lim_{x \to \infty} 5 + lim_{x \to \infty} \frac{1}{x^{2}}}{lim_{x \to \infty} 1}}}$

Step 3.5.5

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

$\frac{1}{\sqrt{5 \frac{5 + lim_{x \to \infty} \frac{1}{x^{2}}}{lim_{x \to \infty} 1}}}$

Step 3.6

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

$\frac{1}{\sqrt{5 \frac{5 + 0}{lim_{x \to \infty} 1}}}$

Step 3.7

Evaluate the limit.

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Step 3.7.1

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

$\frac{1}{\sqrt{5 \frac{5 + 0}{1}}}$

Step 3.7.2

Simplify the answer.

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Step 3.7.2.1

Divide $5 + 0$ by $1$ .

$\frac{1}{\sqrt{5 (5 + 0)}}$

Step 3.7.2.2

Simplify the denominator.

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Step 3.7.2.2.1

Add $5$ and $0$ .

$\frac{1}{\sqrt{5 \cdot 5}}$

Step 3.7.2.2.2

Multiply $5$ by $5$ .

$\frac{1}{\sqrt{25}}$

Step 3.7.2.2.3

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

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

Step 3.7.2.2.4

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

$\frac{1}{5}$

Step 4

Evaluate $lim_{x \to - \infty} \frac{x}{\sqrt{25 x^{2} + 5}}$ to find the horizontal asymptote.

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Step 4.1

Factor $5$ out of $25 x^{2} + 5$ .

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Step 4.1.1

Factor $5$ out of $25 x^{2}$ .

$lim_{x \to - \infty} \frac{x}{\sqrt{5 (5 x^{2}) + 5}}$

Step 4.1.2

Factor $5$ out of $5$ .

$lim_{x \to - \infty} \frac{x}{\sqrt{5 (5 x^{2}) + 5 (1)}}$

Step 4.1.3

Factor $5$ out of $5 (5 x^{2}) + 5 (1)$ .

$lim_{x \to - \infty} \frac{x}{\sqrt{5 (5 x^{2} + 1)}}$

Step 4.2

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

$lim_{x \to - \infty} \frac{\frac{x}{x}}{- \sqrt{\frac{5 (5 x^{2} + 1)}{x^{2}}}}$

Step 4.3

Evaluate the limit.

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Step 4.3.1

Cancel the common factor of $x$ .

$lim_{x \to - \infty} \frac{1}{- \sqrt{\frac{5 (5 x^{2} + 1)}{x^{2}}}}$

Step 4.3.2

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

$\frac{lim_{x \to - \infty} 1}{lim_{x \to - \infty} - \sqrt{\frac{5 (5 x^{2} + 1)}{x^{2}}}}$

Step 4.3.3

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

$\frac{1}{lim_{x \to - \infty} - \sqrt{\frac{5 (5 x^{2} + 1)}{x^{2}}}}$

Step 4.3.4

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{- lim_{x \to - \infty} \sqrt{\frac{5 (5 x^{2} + 1)}{x^{2}}}}$

Step 4.3.5

Move the limit under the radical sign.

$\frac{1}{- \sqrt{lim_{x \to - \infty} \frac{5 (5 x^{2} + 1)}{x^{2}}}}$

Step 4.3.6

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{- \sqrt{5 lim_{x \to - \infty} \frac{5 x^{2} + 1}{x^{2}}}}$

Step 4.4

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

$\frac{1}{- \sqrt{5 lim_{x \to - \infty} \frac{\frac{5 x^{2}}{x^{2}} + \frac{1}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}$

Step 4.5

Evaluate the limit.

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Step 4.5.1

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

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Step 4.5.1.1

Cancel the common factor.

$\frac{1}{- \sqrt{5 lim_{x \to - \infty} \frac{\frac{5 x^{2}}{x^{2}} + \frac{1}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}$

Step 4.5.1.2

Divide $5$ by $1$ .

$\frac{1}{- \sqrt{5 lim_{x \to - \infty} \frac{5 + \frac{1}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}$

Step 4.5.2

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

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Step 4.5.2.1

Cancel the common factor.

$\frac{1}{- \sqrt{5 lim_{x \to - \infty} \frac{5 + \frac{1}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}$

Step 4.5.2.2

Rewrite the expression.

$\frac{1}{- \sqrt{5 lim_{x \to - \infty} \frac{5 + \frac{1}{x^{2}}}{1}}}$

Step 4.5.3

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

$\frac{1}{- \sqrt{5 \frac{lim_{x \to - \infty} 5 + \frac{1}{x^{2}}}{lim_{x \to - \infty} 1}}}$

Step 4.5.4

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

$\frac{1}{- \sqrt{5 \frac{lim_{x \to - \infty} 5 + lim_{x \to - \infty} \frac{1}{x^{2}}}{lim_{x \to - \infty} 1}}}$

Step 4.5.5

Evaluate the limit of $5$ which is constant as $x$ approaches $- \infty$ .

$\frac{1}{- \sqrt{5 \frac{5 + lim_{x \to - \infty} \frac{1}{x^{2}}}{lim_{x \to - \infty} 1}}}$

Step 4.6

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

$\frac{1}{- \sqrt{5 \frac{5 + 0}{lim_{x \to - \infty} 1}}}$

Step 4.7

Evaluate the limit.

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Step 4.7.1

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

$\frac{1}{- \sqrt{5 \frac{5 + 0}{1}}}$

Step 4.7.2

Simplify the answer.

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Step 4.7.2.1

Divide $5 + 0$ by $1$ .

$\frac{1}{- \sqrt{5 (5 + 0)}}$

Step 4.7.2.2

Cancel the common factor of $1$ and $- 1$ .

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Step 4.7.2.2.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{- \sqrt{5 (5 + 0)}}$

Step 4.7.2.2.2

Move the negative in front of the fraction.

$- \frac{1}{\sqrt{5 (5 + 0)}}$

Step 4.7.2.3

Simplify the denominator.

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Step 4.7.2.3.1

Add $5$ and $0$ .

$- \frac{1}{\sqrt{5 \cdot 5}}$

Step 4.7.2.3.2

Multiply $5$ by $5$ .

$- \frac{1}{\sqrt{25}}$

Step 4.7.2.3.3

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

$- \frac{1}{\sqrt{5^{2}}}$

Step 4.7.2.3.4

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

$- \frac{1}{5}$

Step 5

List the horizontal asymptotes:

$y = \frac{1}{5}, - \frac{1}{5}$

Step 6

Use polynomial division to find the oblique asymptotes. Because this expression contains a radical, polynomial division cannot be performed.

Cannot Find Oblique Asymptotes

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

Horizontal Asymptotes: $y = \frac{1}{5}, - \frac{1}{5}$

Cannot Find Oblique Asymptotes

Step 8