Calculus Examples

$\frac{\sqrt{9 x^{2} + 2 x + 5} - 4}{x - 1}$

Step 1

Find where the expression $\frac{\sqrt{9 x^{2} + 2 x + 5} - 4}{x - 1}$ is undefined.

$x = 1$

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

Evaluate $lim_{x \to \infty} \frac{\sqrt{9 x^{2} + 2 x + 5} - 4}{x - 1}$ to find the horizontal asymptote.

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Step 3.1

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

$lim_{x \to \infty} \frac{\sqrt{\frac{9 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} + \frac{5}{x^{2}}} + \frac{- 4}{x}}{\frac{x}{x} - \frac{1}{x}}$

Step 3.2

Evaluate the limit.

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Step 3.2.1

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{\sqrt{\frac{9 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} + \frac{5}{x^{2}}} - \frac{4}{x}}{\frac{x}{x} - \frac{1}{x}}$

Step 3.2.2

Simplify terms.

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Step 3.2.2.1

Cancel the common factor of $x$ .

$lim_{x \to \infty} \frac{\sqrt{\frac{9 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} + \frac{5}{x^{2}}} - \frac{4}{x}}{1 - \frac{1}{x}}$

Step 3.2.2.2

Simplify each term.

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Step 3.2.2.2.1

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

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Step 3.2.2.2.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\sqrt{\frac{9 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} + \frac{5}{x^{2}}} - \frac{4}{x}}{1 - \frac{1}{x}}$

Step 3.2.2.2.1.2

Divide $9$ by $1$ .

$lim_{x \to \infty} \frac{\sqrt{9 + \frac{2 x}{x^{2}} + \frac{5}{x^{2}}} - \frac{4}{x}}{1 - \frac{1}{x}}$

Step 3.2.2.2.2

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

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Step 3.2.2.2.2.1

Factor $x$ out of $2 x$ .

$lim_{x \to \infty} \frac{\sqrt{9 + \frac{x \cdot 2}{x^{2}} + \frac{5}{x^{2}}} - \frac{4}{x}}{1 - \frac{1}{x}}$

Step 3.2.2.2.2.2

Cancel the common factors.

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Step 3.2.2.2.2.2.1

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

$lim_{x \to \infty} \frac{\sqrt{9 + \frac{x \cdot 2}{x \cdot x} + \frac{5}{x^{2}}} - \frac{4}{x}}{1 - \frac{1}{x}}$

Step 3.2.2.2.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\sqrt{9 + \frac{x \cdot 2}{x \cdot x} + \frac{5}{x^{2}}} - \frac{4}{x}}{1 - \frac{1}{x}}$

Step 3.2.2.2.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\sqrt{9 + \frac{2}{x} + \frac{5}{x^{2}}} - \frac{4}{x}}{1 - \frac{1}{x}}$

Step 3.2.3

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

$\frac{lim_{x \to \infty} \sqrt{9 + \frac{2}{x} + \frac{5}{x^{2}}} - \frac{4}{x}}{lim_{x \to \infty} 1 - \frac{1}{x}}$

Step 3.2.4

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

$\frac{lim_{x \to \infty} \sqrt{9 + \frac{2}{x} + \frac{5}{x^{2}}} - lim_{x \to \infty} \frac{4}{x}}{lim_{x \to \infty} 1 - \frac{1}{x}}$

Step 3.2.5

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to \infty} 9 + \frac{2}{x} + \frac{5}{x^{2}}} - lim_{x \to \infty} \frac{4}{x}}{lim_{x \to \infty} 1 - \frac{1}{x}}$

Step 3.2.6

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

$\frac{\sqrt{lim_{x \to \infty} 9 + lim_{x \to \infty} \frac{2}{x} + lim_{x \to \infty} \frac{5}{x^{2}}} - lim_{x \to \infty} \frac{4}{x}}{lim_{x \to \infty} 1 - \frac{1}{x}}$

Step 3.2.7

Evaluate the limit of $9$ which is constant as $x$ approaches $\infty$ .

$\frac{\sqrt{9 + lim_{x \to \infty} \frac{2}{x} + lim_{x \to \infty} \frac{5}{x^{2}}} - lim_{x \to \infty} \frac{4}{x}}{lim_{x \to \infty} 1 - \frac{1}{x}}$

Step 3.2.8

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{9 + 2 lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} \frac{5}{x^{2}}} - lim_{x \to \infty} \frac{4}{x}}{lim_{x \to \infty} 1 - \frac{1}{x}}$

Step 3.3

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

$\frac{\sqrt{9 + 2 \cdot 0 + lim_{x \to \infty} \frac{5}{x^{2}}} - lim_{x \to \infty} \frac{4}{x}}{lim_{x \to \infty} 1 - \frac{1}{x}}$

Step 3.4

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

$\frac{\sqrt{9 + 2 \cdot 0 + 5 lim_{x \to \infty} \frac{1}{x^{2}}} - lim_{x \to \infty} \frac{4}{x}}{lim_{x \to \infty} 1 - \frac{1}{x}}$

Step 3.5

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

$\frac{\sqrt{9 + 2 \cdot 0 + 5 \cdot 0} - lim_{x \to \infty} \frac{4}{x}}{lim_{x \to \infty} 1 - \frac{1}{x}}$

Step 3.6

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

$\frac{\sqrt{9 + 2 \cdot 0 + 5 \cdot 0} - 4 lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 1 - \frac{1}{x}}$

Step 3.7

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

$\frac{\sqrt{9 + 2 \cdot 0 + 5 \cdot 0} - 4 \cdot 0}{lim_{x \to \infty} 1 - \frac{1}{x}}$

Step 3.8

Evaluate the limit.

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Step 3.8.1

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

$\frac{\sqrt{9 + 2 \cdot 0 + 5 \cdot 0} - 4 \cdot 0}{lim_{x \to \infty} 1 - lim_{x \to \infty} \frac{1}{x}}$

Step 3.8.2

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

$\frac{\sqrt{9 + 2 \cdot 0 + 5 \cdot 0} - 4 \cdot 0}{1 - lim_{x \to \infty} \frac{1}{x}}$

Step 3.9

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

$\frac{\sqrt{9 + 2 \cdot 0 + 5 \cdot 0} - 4 \cdot 0}{1 - 0}$

Step 3.10

Simplify the answer.

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Step 3.10.1

Simplify the numerator.

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Step 3.10.1.1

Multiply $2$ by $0$ .

$\frac{\sqrt{9 + 0 + 5 \cdot 0} - 4 \cdot 0}{1 - 0}$

Step 3.10.1.2

Multiply $5$ by $0$ .

$\frac{\sqrt{9 + 0 + 0} - 4 \cdot 0}{1 - 0}$

Step 3.10.1.3

Add $9 + 0$ and $0$ .

$\frac{\sqrt{9 + 0} - 4 \cdot 0}{1 - 0}$

Step 3.10.1.4

Add $9$ and $0$ .

$\frac{\sqrt{9} - 4 \cdot 0}{1 - 0}$

Step 3.10.1.5

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2}} - 4 \cdot 0}{1 - 0}$

Step 3.10.1.6

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

$\frac{3 - 4 \cdot 0}{1 - 0}$

Step 3.10.1.7

Multiply $- 4$ by $0$ .

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

Step 3.10.1.8

Add $3$ and $0$ .

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

Step 3.10.2

Simplify the denominator.

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Step 3.10.2.1

Multiply $- 1$ by $0$ .

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

Step 3.10.2.2

Add $1$ and $0$ .

$\frac{3}{1}$

Step 3.10.3

Divide $3$ by $1$ .

$3$

Step 4

Evaluate $lim_{x \to - \infty} \frac{\sqrt{9 x^{2} + 2 x + 5} - 4}{x - 1}$ to find the horizontal asymptote.

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Step 4.1

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

$lim_{x \to - \infty} \frac{- \sqrt{\frac{9 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} + \frac{5}{x^{2}}} + \frac{- 4}{x}}{\frac{x}{x} - \frac{1}{x}}$

Step 4.2

Evaluate the limit.

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Step 4.2.1

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{- \sqrt{\frac{9 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} + \frac{5}{x^{2}}} - \frac{4}{x}}{\frac{x}{x} - \frac{1}{x}}$

Step 4.2.2

Simplify terms.

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Step 4.2.2.1

Cancel the common factor of $x$ .

$lim_{x \to - \infty} \frac{- \sqrt{\frac{9 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} + \frac{5}{x^{2}}} - \frac{4}{x}}{1 - \frac{1}{x}}$

Step 4.2.2.2

Simplify each term.

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Step 4.2.2.2.1

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

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Step 4.2.2.2.1.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{- \sqrt{\frac{9 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} + \frac{5}{x^{2}}} - \frac{4}{x}}{1 - \frac{1}{x}}$

Step 4.2.2.2.1.2

Divide $9$ by $1$ .

$lim_{x \to - \infty} \frac{- \sqrt{9 + \frac{2 x}{x^{2}} + \frac{5}{x^{2}}} - \frac{4}{x}}{1 - \frac{1}{x}}$

Step 4.2.2.2.2

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

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Step 4.2.2.2.2.1

Factor $x$ out of $2 x$ .

$lim_{x \to - \infty} \frac{- \sqrt{9 + \frac{x \cdot 2}{x^{2}} + \frac{5}{x^{2}}} - \frac{4}{x}}{1 - \frac{1}{x}}$

Step 4.2.2.2.2.2

Cancel the common factors.

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Step 4.2.2.2.2.2.1

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

$lim_{x \to - \infty} \frac{- \sqrt{9 + \frac{x \cdot 2}{x \cdot x} + \frac{5}{x^{2}}} - \frac{4}{x}}{1 - \frac{1}{x}}$

Step 4.2.2.2.2.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{- \sqrt{9 + \frac{x \cdot 2}{x \cdot x} + \frac{5}{x^{2}}} - \frac{4}{x}}{1 - \frac{1}{x}}$

Step 4.2.2.2.2.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{- \sqrt{9 + \frac{2}{x} + \frac{5}{x^{2}}} - \frac{4}{x}}{1 - \frac{1}{x}}$

Step 4.2.3

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

$\frac{lim_{x \to - \infty} - \sqrt{9 + \frac{2}{x} + \frac{5}{x^{2}}} - \frac{4}{x}}{lim_{x \to - \infty} 1 - \frac{1}{x}}$

Step 4.2.4

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

$\frac{- lim_{x \to - \infty} \sqrt{9 + \frac{2}{x} + \frac{5}{x^{2}}} - lim_{x \to - \infty} \frac{4}{x}}{lim_{x \to - \infty} 1 - \frac{1}{x}}$

Step 4.2.5

Move the limit under the radical sign.

$\frac{- \sqrt{lim_{x \to - \infty} 9 + \frac{2}{x} + \frac{5}{x^{2}}} - lim_{x \to - \infty} \frac{4}{x}}{lim_{x \to - \infty} 1 - \frac{1}{x}}$

Step 4.2.6

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

$\frac{- \sqrt{lim_{x \to - \infty} 9 + lim_{x \to - \infty} \frac{2}{x} + lim_{x \to - \infty} \frac{5}{x^{2}}} - lim_{x \to - \infty} \frac{4}{x}}{lim_{x \to - \infty} 1 - \frac{1}{x}}$

Step 4.2.7

Evaluate the limit of $9$ which is constant as $x$ approaches $- \infty$ .

$\frac{- \sqrt{9 + lim_{x \to - \infty} \frac{2}{x} + lim_{x \to - \infty} \frac{5}{x^{2}}} - lim_{x \to - \infty} \frac{4}{x}}{lim_{x \to - \infty} 1 - \frac{1}{x}}$

Step 4.2.8

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{- \sqrt{9 + 2 lim_{x \to - \infty} \frac{1}{x} + lim_{x \to - \infty} \frac{5}{x^{2}}} - lim_{x \to - \infty} \frac{4}{x}}{lim_{x \to - \infty} 1 - \frac{1}{x}}$

Step 4.3

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

$\frac{- \sqrt{9 + 2 \cdot 0 + lim_{x \to - \infty} \frac{5}{x^{2}}} - lim_{x \to - \infty} \frac{4}{x}}{lim_{x \to - \infty} 1 - \frac{1}{x}}$

Step 4.4

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

$\frac{- \sqrt{9 + 2 \cdot 0 + 5 lim_{x \to - \infty} \frac{1}{x^{2}}} - lim_{x \to - \infty} \frac{4}{x}}{lim_{x \to - \infty} 1 - \frac{1}{x}}$

Step 4.5

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

$\frac{- \sqrt{9 + 2 \cdot 0 + 5 \cdot 0} - lim_{x \to - \infty} \frac{4}{x}}{lim_{x \to - \infty} 1 - \frac{1}{x}}$

Step 4.6

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

$\frac{- \sqrt{9 + 2 \cdot 0 + 5 \cdot 0} - 4 lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} 1 - \frac{1}{x}}$

Step 4.7

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

$\frac{- \sqrt{9 + 2 \cdot 0 + 5 \cdot 0} - 4 \cdot 0}{lim_{x \to - \infty} 1 - \frac{1}{x}}$

Step 4.8

Evaluate the limit.

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Step 4.8.1

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

$\frac{- \sqrt{9 + 2 \cdot 0 + 5 \cdot 0} - 4 \cdot 0}{lim_{x \to - \infty} 1 - lim_{x \to - \infty} \frac{1}{x}}$

Step 4.8.2

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

$\frac{- \sqrt{9 + 2 \cdot 0 + 5 \cdot 0} - 4 \cdot 0}{1 - lim_{x \to - \infty} \frac{1}{x}}$

Step 4.9

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

$\frac{- \sqrt{9 + 2 \cdot 0 + 5 \cdot 0} - 4 \cdot 0}{1 - 0}$

Step 4.10

Simplify the answer.

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Step 4.10.1

Simplify the numerator.

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Step 4.10.1.1

Multiply $2$ by $0$ .

$\frac{- \sqrt{9 + 0 + 5 \cdot 0} - 4 \cdot 0}{1 - 0}$

Step 4.10.1.2

Multiply $5$ by $0$ .

$\frac{- \sqrt{9 + 0 + 0} - 4 \cdot 0}{1 - 0}$

Step 4.10.1.3

Add $9 + 0$ and $0$ .

$\frac{- \sqrt{9 + 0} - 4 \cdot 0}{1 - 0}$

Step 4.10.1.4

Add $9$ and $0$ .

$\frac{- \sqrt{9} - 4 \cdot 0}{1 - 0}$

Step 4.10.1.5

Rewrite $9$ as $3^{2}$ .

$\frac{- \sqrt{3^{2}} - 4 \cdot 0}{1 - 0}$

Step 4.10.1.6

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

$\frac{- 1 \cdot 3 - 4 \cdot 0}{1 - 0}$

Step 4.10.1.7

Multiply $- 1$ by $3$ .

$\frac{- 3 - 4 \cdot 0}{1 - 0}$

Step 4.10.1.8

Multiply $- 4$ by $0$ .

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

Step 4.10.1.9

Add $- 3$ and $0$ .

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

Step 4.10.2

Simplify the denominator.

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Step 4.10.2.1

Multiply $- 1$ by $0$ .

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

Step 4.10.2.2

Add $1$ and $0$ .

$\frac{- 3}{1}$

Step 4.10.3

Divide $- 3$ by $1$ .

$- 3$

Step 5

List the horizontal asymptotes:

$y = 3, - 3$

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: $y = 3, - 3$

Cannot Find Oblique Asymptotes

Step 8