Calculus Examples

$f (x) = \frac{\sqrt{x^{2} + 1}}{x}$

Step 1

Find where the expression $\frac{\sqrt{x^{2} + 1}}{x}$ is undefined.

$x = 0$

Step 2

Evaluate $lim_{x \to \infty} \frac{\sqrt{x^{2} + 1}}{x}$ to find the horizontal asymptote.

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Step 2.1

Evaluate the limit.

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Step 2.1.1

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

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Step 2.1.1.1

Cancel the common factor.

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

Step 2.1.1.2

Rewrite the expression.

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

Step 2.1.2

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

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

Step 2.1.3

Move the limit under the radical sign.

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

Step 2.1.4

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

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

Step 2.1.5

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

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

Step 2.2

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

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

Step 2.3

Evaluate the limit.

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Step 2.3.1

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

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

Step 2.3.2

Simplify the answer.

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Step 2.3.2.1

Divide $\sqrt{1 + 0}$ by $1$ .

$\sqrt{1 + 0}$

Step 2.3.2.2

Add $1$ and $0$ .

$\sqrt{1}$

Step 2.3.2.3

Any root of $1$ is $1$ .

$1$

Step 3

Evaluate $lim_{x \to - \infty} \frac{\sqrt{x^{2} + 1}}{x}$ to find the horizontal asymptote.

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Step 3.1

Evaluate the limit.

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Step 3.1.1

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

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Step 3.1.1.1

Cancel the common factor.

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

Step 3.1.1.2

Rewrite the expression.

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

Move the limit under the radical sign.

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

Step 3.1.5

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

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

Step 3.1.6

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

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

Step 3.2

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

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

Step 3.3

Evaluate the limit.

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Step 3.3.1

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

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

Step 3.3.2

Simplify the answer.

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Step 3.3.2.1

Divide $- \sqrt{1 + 0}$ by $1$ .

$- \sqrt{1 + 0}$

Step 3.3.2.2

Add $1$ and $0$ .

$- \sqrt{1}$

Step 3.3.2.3

Any root of $1$ is $1$ .

$- 1 \cdot 1$

Step 3.3.2.4

Multiply $- 1$ by $1$ .

$- 1$

Step 4

List the horizontal asymptotes:

$y = 1, - 1$

Step 5

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

Cannot Find Oblique Asymptotes

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 1, - 1$

Cannot Find Oblique Asymptotes

Step 7