Precalculus Examples

$f (x) = \frac{- 3 x + 1}{\sqrt{x^{2} + x}}$

Step 1

Find where the expression $\frac{- 3 x + 1}{\sqrt{x^{2} + x}}$ is undefined.

$- 1 \leq x \leq 0$

Step 2

Since $\frac{- 3 x + 1}{\sqrt{x^{2} + x}}$ $\to$ $\infty$ as $x$ $\to$ $- 1$ from the left and $\frac{- 3 x + 1}{\sqrt{x^{2} + x}}$ $\to$ $\infty$ as $x$ $\to$ $- 1$ from the right, then $x = - 1$ is a vertical asymptote.

$x = - 1$

Step 3

Since $\frac{- 3 x + 1}{\sqrt{x^{2} + x}}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the left and $\frac{- 3 x + 1}{\sqrt{x^{2} + x}}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the right, then $x = 0$ is a vertical asymptote.

$x = 0$

Step 4

List all of the vertical asymptotes:

$x = - 1, 0$

Step 5

Evaluate $lim_{x \to \infty} \frac{- 3 x + 1}{\sqrt{x^{2} + x}}$ to find the horizontal asymptote.

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Step 5.1

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

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Step 5.1.1

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

$lim_{x \to \infty} \frac{- 3 x + 1}{\sqrt{x \cdot x + x}}$

Step 5.1.2

Raise $x$ to the power of $1$ .

$lim_{x \to \infty} \frac{- 3 x + 1}{\sqrt{x \cdot x + x^{1}}}$

Step 5.1.3

Factor $x$ out of $x^{1}$ .

$lim_{x \to \infty} \frac{- 3 x + 1}{\sqrt{x \cdot x + x \cdot 1}}$

Step 5.1.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

$lim_{x \to \infty} \frac{- 3 x + 1}{\sqrt{x (x + 1)}}$

Step 5.2

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

$lim_{x \to \infty} \frac{\frac{- 3 x}{x} + \frac{1}{x}}{\sqrt{\frac{x (x + 1)}{x^{2}}}}$

Step 5.3

Cancel the common factor of $x$ .

$lim_{x \to \infty} \frac{- 3 + \frac{1}{x}}{\sqrt{\frac{x (x + 1)}{x^{2}}}}$

Step 5.4

Cancel the common factors.

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Step 5.4.1

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

$lim_{x \to \infty} \frac{- 3 + \frac{1}{x}}{\sqrt{\frac{x (x + 1)}{x \cdot x}}}$

Step 5.4.2

Cancel the common factor.

$lim_{x \to \infty} \frac{- 3 + \frac{1}{x}}{\sqrt{\frac{x (x + 1)}{x \cdot x}}}$

Step 5.4.3

Rewrite the expression.

$lim_{x \to \infty} \frac{- 3 + \frac{1}{x}}{\sqrt{\frac{x + 1}{x}}}$

Step 5.5

Evaluate the limit.

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Step 5.5.1

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

$\frac{lim_{x \to \infty} - 3 + \frac{1}{x}}{lim_{x \to \infty} \sqrt{\frac{x + 1}{x}}}$

Step 5.5.2

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

$\frac{- lim_{x \to \infty} 3 + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} \sqrt{\frac{x + 1}{x}}}$

Step 5.5.3

Evaluate the limit of $3$ which is constant as $x$ approaches $\infty$ .

$\frac{- 1 \cdot 3 + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} \sqrt{\frac{x + 1}{x}}}$

Step 5.6

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

$\frac{- 3 + 0}{lim_{x \to \infty} \sqrt{\frac{x + 1}{x}}}$

Step 5.7

Move the limit under the radical sign.

$\frac{- 3 + 0}{\sqrt{lim_{x \to \infty} \frac{x + 1}{x}}}$

Step 5.8

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

$\frac{- 3 + 0}{\sqrt{lim_{x \to \infty} \frac{\frac{x}{x} + \frac{1}{x}}{\frac{x}{x}}}}$

Step 5.9

Evaluate the limit.

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Step 5.9.1

Cancel the common factor of $x$ .

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Step 5.9.1.1

Cancel the common factor.

$\frac{- 3 + 0}{\sqrt{lim_{x \to \infty} \frac{\frac{x}{x} + \frac{1}{x}}{\frac{x}{x}}}}$

Step 5.9.1.2

Rewrite the expression.

$\frac{- 3 + 0}{\sqrt{lim_{x \to \infty} \frac{1 + \frac{1}{x}}{\frac{x}{x}}}}$

Step 5.9.2

Cancel the common factor of $x$ .

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Step 5.9.2.1

Cancel the common factor.

$\frac{- 3 + 0}{\sqrt{lim_{x \to \infty} \frac{1 + \frac{1}{x}}{\frac{x}{x}}}}$

Step 5.9.2.2

Rewrite the expression.

$\frac{- 3 + 0}{\sqrt{lim_{x \to \infty} \frac{1 + \frac{1}{x}}{1}}}$

Step 5.9.3

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

$\frac{- 3 + 0}{\sqrt{\frac{lim_{x \to \infty} 1 + \frac{1}{x}}{lim_{x \to \infty} 1}}}$

Step 5.9.4

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

$\frac{- 3 + 0}{\sqrt{\frac{lim_{x \to \infty} 1 + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 1}}}$

Step 5.9.5

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

$\frac{- 3 + 0}{\sqrt{\frac{1 + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 1}}}$

Step 5.10

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

$\frac{- 3 + 0}{\sqrt{\frac{1 + 0}{lim_{x \to \infty} 1}}}$

Step 5.11

Evaluate the limit.

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Step 5.11.1

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

$\frac{- 3 + 0}{\sqrt{\frac{1 + 0}{1}}}$

Step 5.11.2

Simplify the answer.

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Step 5.11.2.1

Divide $1 + 0$ by $1$ .

$\frac{- 3 + 0}{\sqrt{1 + 0}}$

Step 5.11.2.2

Add $- 3$ and $0$ .

$\frac{- 3}{\sqrt{1 + 0}}$

Step 5.11.2.3

Simplify the denominator.

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Step 5.11.2.3.1

Add $1$ and $0$ .

$\frac{- 3}{\sqrt{1}}$

Step 5.11.2.3.2

Any root of $1$ is $1$ .

$\frac{- 3}{1}$

Step 5.11.2.4

Divide $- 3$ by $1$ .

$- 3$

Step 6

Evaluate $lim_{x \to - \infty} \frac{- 3 x + 1}{\sqrt{x^{2} + x}}$ to find the horizontal asymptote.

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Step 6.1

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

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Step 6.1.1

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

$lim_{x \to - \infty} \frac{- 3 x + 1}{\sqrt{x \cdot x + x}}$

Step 6.1.2

Raise $x$ to the power of $1$ .

$lim_{x \to - \infty} \frac{- 3 x + 1}{\sqrt{x \cdot x + x^{1}}}$

Step 6.1.3

Factor $x$ out of $x^{1}$ .

$lim_{x \to - \infty} \frac{- 3 x + 1}{\sqrt{x \cdot x + x \cdot 1}}$

Step 6.1.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

$lim_{x \to - \infty} \frac{- 3 x + 1}{\sqrt{x (x + 1)}}$

Step 6.2

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

$lim_{x \to - \infty} \frac{\frac{- 3 x}{x} + \frac{1}{x}}{- \sqrt{\frac{x (x + 1)}{x^{2}}}}$

Step 6.3

Cancel the common factor of $x$ .

$lim_{x \to - \infty} \frac{- 3 + \frac{1}{x}}{- \sqrt{\frac{x (x + 1)}{x^{2}}}}$

Step 6.4

Cancel the common factors.

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Step 6.4.1

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

$lim_{x \to - \infty} \frac{- 3 + \frac{1}{x}}{- \sqrt{\frac{x (x + 1)}{x \cdot x}}}$

Step 6.4.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{- 3 + \frac{1}{x}}{- \sqrt{\frac{x (x + 1)}{x \cdot x}}}$

Step 6.4.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{- 3 + \frac{1}{x}}{- \sqrt{\frac{x + 1}{x}}}$

Step 6.5

Evaluate the limit.

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Step 6.5.1

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

$\frac{lim_{x \to - \infty} - 3 + \frac{1}{x}}{lim_{x \to - \infty} - \sqrt{\frac{x + 1}{x}}}$

Step 6.5.2

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

$\frac{- lim_{x \to - \infty} 3 + lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} - \sqrt{\frac{x + 1}{x}}}$

Step 6.5.3

Evaluate the limit of $3$ which is constant as $x$ approaches $- \infty$ .

$\frac{- 1 \cdot 3 + lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} - \sqrt{\frac{x + 1}{x}}}$

Step 6.6

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

$\frac{- 3 + 0}{lim_{x \to - \infty} - \sqrt{\frac{x + 1}{x}}}$

Step 6.7

Evaluate the limit.

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Step 6.7.1

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

$\frac{- 3 + 0}{- lim_{x \to - \infty} \sqrt{\frac{x + 1}{x}}}$

Step 6.7.2

Move the limit under the radical sign.

$\frac{- 3 + 0}{- \sqrt{lim_{x \to - \infty} \frac{x + 1}{x}}}$

Step 6.8

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

$\frac{- 3 + 0}{- \sqrt{lim_{x \to - \infty} \frac{\frac{x}{x} + \frac{1}{x}}{\frac{x}{x}}}}$

Step 6.9

Evaluate the limit.

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Step 6.9.1

Cancel the common factor of $x$ .

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Step 6.9.1.1

Cancel the common factor.

$\frac{- 3 + 0}{- \sqrt{lim_{x \to - \infty} \frac{\frac{x}{x} + \frac{1}{x}}{\frac{x}{x}}}}$

Step 6.9.1.2

Rewrite the expression.

$\frac{- 3 + 0}{- \sqrt{lim_{x \to - \infty} \frac{1 + \frac{1}{x}}{\frac{x}{x}}}}$

Step 6.9.2

Cancel the common factor of $x$ .

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Step 6.9.2.1

Cancel the common factor.

$\frac{- 3 + 0}{- \sqrt{lim_{x \to - \infty} \frac{1 + \frac{1}{x}}{\frac{x}{x}}}}$

Step 6.9.2.2

Rewrite the expression.

$\frac{- 3 + 0}{- \sqrt{lim_{x \to - \infty} \frac{1 + \frac{1}{x}}{1}}}$

Step 6.9.3

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

$\frac{- 3 + 0}{- \sqrt{\frac{lim_{x \to - \infty} 1 + \frac{1}{x}}{lim_{x \to - \infty} 1}}}$

Step 6.9.4

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

$\frac{- 3 + 0}{- \sqrt{\frac{lim_{x \to - \infty} 1 + lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} 1}}}$

Step 6.9.5

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

$\frac{- 3 + 0}{- \sqrt{\frac{1 + lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} 1}}}$

Step 6.10

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

$\frac{- 3 + 0}{- \sqrt{\frac{1 + 0}{lim_{x \to - \infty} 1}}}$

Step 6.11

Evaluate the limit.

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Step 6.11.1

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

$\frac{- 3 + 0}{- \sqrt{\frac{1 + 0}{1}}}$

Step 6.11.2

Simplify the answer.

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Step 6.11.2.1

Divide $1 + 0$ by $1$ .

$\frac{- 3 + 0}{- \sqrt{1 + 0}}$

Step 6.11.2.2

Add $- 3$ and $0$ .

$\frac{- 3}{- \sqrt{1 + 0}}$

Step 6.11.2.3

Simplify the denominator.

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Step 6.11.2.3.1

Add $1$ and $0$ .

$\frac{- 3}{- \sqrt{1}}$

Step 6.11.2.3.2

Any root of $1$ is $1$ .

$\frac{- 3}{- 1 \cdot 1}$

Step 6.11.2.4

Multiply $- 1$ by $1$ .

$\frac{- 3}{- 1}$

Step 6.11.2.5

Divide $- 3$ by $- 1$ .

$3$

Step 7

List the horizontal asymptotes:

$y = - 3, 3$

Step 8

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

Cannot Find Oblique Asymptotes

Step 9

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 1, 0$

Horizontal Asymptotes: $y = - 3, 3$

Cannot Find Oblique Asymptotes

Step 10