Calculus Examples

$f (x) = \frac{\sqrt{10 x^{2} + 11}}{12 x + 10}$

Step 1

Find where the expression $\frac{\sqrt{10 x^{2} + 11}}{12 x + 10}$ is undefined.

$x = - \frac{5}{6}$

Step 2

Evaluate $lim_{x \to \infty} \frac{\sqrt{10 x^{2} + 11}}{12 x + 10}$ to find the horizontal asymptote.

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Step 2.1

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

$lim_{x \to \infty} \frac{\sqrt{\frac{10 x^{2}}{x^{2}} + \frac{11}{x^{2}}}}{\frac{12 x}{x} + \frac{10}{x}}$

Step 2.2

Evaluate the limit.

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Step 2.2.1

Cancel the common factor of $x$ .

$lim_{x \to \infty} \frac{\sqrt{\frac{10 x^{2}}{x^{2}} + \frac{11}{x^{2}}}}{12 + \frac{10}{x}}$

Step 2.2.2

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

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Step 2.2.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\sqrt{\frac{10 x^{2}}{x^{2}} + \frac{11}{x^{2}}}}{12 + \frac{10}{x}}$

Step 2.2.2.2

Divide $10$ by $1$ .

$lim_{x \to \infty} \frac{\sqrt{10 + \frac{11}{x^{2}}}}{12 + \frac{10}{x}}$

Step 2.2.3

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

$\frac{lim_{x \to \infty} \sqrt{10 + \frac{11}{x^{2}}}}{lim_{x \to \infty} 12 + \frac{10}{x}}$

Step 2.2.4

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to \infty} 10 + \frac{11}{x^{2}}}}{lim_{x \to \infty} 12 + \frac{10}{x}}$

Step 2.2.5

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

$\frac{\sqrt{lim_{x \to \infty} 10 + lim_{x \to \infty} \frac{11}{x^{2}}}}{lim_{x \to \infty} 12 + \frac{10}{x}}$

Step 2.2.6

Evaluate the limit of $10$ which is constant as $x$ approaches $\infty$ .

$\frac{\sqrt{10 + lim_{x \to \infty} \frac{11}{x^{2}}}}{lim_{x \to \infty} 12 + \frac{10}{x}}$

Step 2.2.7

Move the term $11$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{10 + 11 lim_{x \to \infty} \frac{1}{x^{2}}}}{lim_{x \to \infty} 12 + \frac{10}{x}}$

Step 2.3

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

$\frac{\sqrt{10 + 11 \cdot 0}}{lim_{x \to \infty} 12 + \frac{10}{x}}$

Step 2.4

Evaluate the limit.

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Step 2.4.1

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

$\frac{\sqrt{10 + 11 \cdot 0}}{lim_{x \to \infty} 12 + lim_{x \to \infty} \frac{10}{x}}$

Step 2.4.2

Evaluate the limit of $12$ which is constant as $x$ approaches $\infty$ .

$\frac{\sqrt{10 + 11 \cdot 0}}{12 + lim_{x \to \infty} \frac{10}{x}}$

Step 2.4.3

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{10 + 11 \cdot 0}}{12 + 10 lim_{x \to \infty} \frac{1}{x}}$

Step 2.5

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

$\frac{\sqrt{10 + 11 \cdot 0}}{12 + 10 \cdot 0}$

Step 2.6

Simplify the answer.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Multiply $11$ by $0$ .

$\frac{\sqrt{10 + 0}}{12 + 10 \cdot 0}$

Step 2.6.1.2

Add $10$ and $0$ .

$\frac{\sqrt{10}}{12 + 10 \cdot 0}$

Step 2.6.2

Simplify the denominator.

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Step 2.6.2.1

Multiply $10$ by $0$ .

$\frac{\sqrt{10}}{12 + 0}$

Step 2.6.2.2

Add $12$ and $0$ .

$\frac{\sqrt{10}}{12}$

Step 3

Evaluate $lim_{x \to - \infty} \frac{\sqrt{10 x^{2} + 11}}{12 x + 10}$ 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 $- \sqrt{x^{2}} = x$ .

$lim_{x \to - \infty} \frac{- \sqrt{\frac{10 x^{2}}{x^{2}} + \frac{11}{x^{2}}}}{\frac{12 x}{x} + \frac{10}{x}}$

Step 3.2

Evaluate the limit.

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Step 3.2.1

Cancel the common factor of $x$ .

$lim_{x \to - \infty} \frac{- \sqrt{\frac{10 x^{2}}{x^{2}} + \frac{11}{x^{2}}}}{12 + \frac{10}{x}}$

Step 3.2.2

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

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Step 3.2.2.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{- \sqrt{\frac{10 x^{2}}{x^{2}} + \frac{11}{x^{2}}}}{12 + \frac{10}{x}}$

Step 3.2.2.2

Divide $10$ by $1$ .

$lim_{x \to - \infty} \frac{- \sqrt{10 + \frac{11}{x^{2}}}}{12 + \frac{10}{x}}$

Step 3.2.3

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

$\frac{lim_{x \to - \infty} - \sqrt{10 + \frac{11}{x^{2}}}}{lim_{x \to - \infty} 12 + \frac{10}{x}}$

Step 3.2.4

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

$\frac{- lim_{x \to - \infty} \sqrt{10 + \frac{11}{x^{2}}}}{lim_{x \to - \infty} 12 + \frac{10}{x}}$

Step 3.2.5

Move the limit under the radical sign.

$\frac{- \sqrt{lim_{x \to - \infty} 10 + \frac{11}{x^{2}}}}{lim_{x \to - \infty} 12 + \frac{10}{x}}$

Step 3.2.6

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

$\frac{- \sqrt{lim_{x \to - \infty} 10 + lim_{x \to - \infty} \frac{11}{x^{2}}}}{lim_{x \to - \infty} 12 + \frac{10}{x}}$

Step 3.2.7

Evaluate the limit of $10$ which is constant as $x$ approaches $- \infty$ .

$\frac{- \sqrt{10 + lim_{x \to - \infty} \frac{11}{x^{2}}}}{lim_{x \to - \infty} 12 + \frac{10}{x}}$

Step 3.2.8

Move the term $11$ outside of the limit because it is constant with respect to $x$ .

$\frac{- \sqrt{10 + 11 lim_{x \to - \infty} \frac{1}{x^{2}}}}{lim_{x \to - \infty} 12 + \frac{10}{x}}$

Step 3.3

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

$\frac{- \sqrt{10 + 11 \cdot 0}}{lim_{x \to - \infty} 12 + \frac{10}{x}}$

Step 3.4

Evaluate the limit.

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Step 3.4.1

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

$\frac{- \sqrt{10 + 11 \cdot 0}}{lim_{x \to - \infty} 12 + lim_{x \to - \infty} \frac{10}{x}}$

Step 3.4.2

Evaluate the limit of $12$ which is constant as $x$ approaches $- \infty$ .

$\frac{- \sqrt{10 + 11 \cdot 0}}{12 + lim_{x \to - \infty} \frac{10}{x}}$

Step 3.4.3

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$\frac{- \sqrt{10 + 11 \cdot 0}}{12 + 10 lim_{x \to - \infty} \frac{1}{x}}$

Step 3.5

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

$\frac{- \sqrt{10 + 11 \cdot 0}}{12 + 10 \cdot 0}$

Step 3.6

Simplify the answer.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Multiply $11$ by $0$ .

$\frac{- \sqrt{10 + 0}}{12 + 10 \cdot 0}$

Step 3.6.1.2

Add $10$ and $0$ .

$\frac{- \sqrt{10}}{12 + 10 \cdot 0}$

Step 3.6.2

Simplify the denominator.

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Step 3.6.2.1

Multiply $10$ by $0$ .

$\frac{- \sqrt{10}}{12 + 0}$

Step 3.6.2.2

Add $12$ and $0$ .

$\frac{- \sqrt{10}}{12}$

Step 3.6.3

Move the negative in front of the fraction.

$- \frac{\sqrt{10}}{12}$

Step 4

List the horizontal asymptotes:

$y = \frac{\sqrt{10}}{12}, - \frac{\sqrt{10}}{12}$

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: $x = - \frac{5}{6}$

Horizontal Asymptotes: $y = \frac{\sqrt{10}}{12}, - \frac{\sqrt{10}}{12}$

Cannot Find Oblique Asymptotes

Step 7