Calculus Examples

$\frac{\sqrt{2 x^{2} - x + 10}}{7 x - 5}$

Step 1

Find where the expression $\frac{\sqrt{2 x^{2} - x + 10}}{7 x - 5}$ is undefined.

$x = \frac{5}{7}$

Step 2

Evaluate $lim_{x \to \infty} \frac{\sqrt{2 x^{2} - x + 10}}{7 x - 5}$ 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 $\sqrt{x^{2}} = x$ .

$lim_{x \to \infty} \frac{\sqrt{\frac{2 x^{2}}{x^{2}} + \frac{- x}{x^{2}} + \frac{10}{x^{2}}}}{\frac{7 x}{x} + \frac{- 5}{x}}$

Step 2.2

Evaluate the limit.

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Step 2.2.1

Simplify each term.

$lim_{x \to \infty} \frac{\sqrt{\frac{2 x^{2}}{x^{2}} + \frac{- x}{x^{2}} + \frac{10}{x^{2}}}}{7 - \frac{5}{x}}$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

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

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Step 2.2.2.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\sqrt{\frac{2 x^{2}}{x^{2}} + \frac{- x}{x^{2}} + \frac{10}{x^{2}}}}{7 - \frac{5}{x}}$

Step 2.2.2.1.2

Divide $2$ by $1$ .

$lim_{x \to \infty} \frac{\sqrt{2 + \frac{- x}{x^{2}} + \frac{10}{x^{2}}}}{7 - \frac{5}{x}}$

Step 2.2.2.2

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

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Step 2.2.2.2.1

Factor $x$ out of $- x$ .

$lim_{x \to \infty} \frac{\sqrt{2 + \frac{x \cdot - 1}{x^{2}} + \frac{10}{x^{2}}}}{7 - \frac{5}{x}}$

Step 2.2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.2.1

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

$lim_{x \to \infty} \frac{\sqrt{2 + \frac{x \cdot - 1}{x \cdot x} + \frac{10}{x^{2}}}}{7 - \frac{5}{x}}$

Step 2.2.2.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\sqrt{2 + \frac{x \cdot - 1}{x \cdot x} + \frac{10}{x^{2}}}}{7 - \frac{5}{x}}$

Step 2.2.2.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\sqrt{2 + \frac{- 1}{x} + \frac{10}{x^{2}}}}{7 - \frac{5}{x}}$

Step 2.2.2.3

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{\sqrt{2 - \frac{1}{x} + \frac{10}{x^{2}}}}{7 - \frac{5}{x}}$

Step 2.2.3

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

$\frac{lim_{x \to \infty} \sqrt{2 - \frac{1}{x} + \frac{10}{x^{2}}}}{lim_{x \to \infty} 7 - \frac{5}{x}}$

Step 2.2.4

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to \infty} 2 - \frac{1}{x} + \frac{10}{x^{2}}}}{lim_{x \to \infty} 7 - \frac{5}{x}}$

Step 2.2.5

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

$\frac{\sqrt{lim_{x \to \infty} 2 - lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} \frac{10}{x^{2}}}}{lim_{x \to \infty} 7 - \frac{5}{x}}$

Step 2.2.6

Evaluate the limit of $2$ which is constant as $x$ approaches $\infty$ .

$\frac{\sqrt{2 - lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} \frac{10}{x^{2}}}}{lim_{x \to \infty} 7 - \frac{5}{x}}$

Step 2.3

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

$\frac{\sqrt{2 - 0 + lim_{x \to \infty} \frac{10}{x^{2}}}}{lim_{x \to \infty} 7 - \frac{5}{x}}$

Step 2.4

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

$\frac{\sqrt{2 - 0 + 10 lim_{x \to \infty} \frac{1}{x^{2}}}}{lim_{x \to \infty} 7 - \frac{5}{x}}$

Step 2.5

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

$\frac{\sqrt{2 - 0 + 10 \cdot 0}}{lim_{x \to \infty} 7 - \frac{5}{x}}$

Step 2.6

Evaluate the limit.

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Step 2.6.1

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

$\frac{\sqrt{2 - 0 + 10 \cdot 0}}{lim_{x \to \infty} 7 - lim_{x \to \infty} \frac{5}{x}}$

Step 2.6.2

Evaluate the limit of $7$ which is constant as $x$ approaches $\infty$ .

$\frac{\sqrt{2 - 0 + 10 \cdot 0}}{7 - lim_{x \to \infty} \frac{5}{x}}$

Step 2.6.3

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

$\frac{\sqrt{2 - 0 + 10 \cdot 0}}{7 - 5 lim_{x \to \infty} \frac{1}{x}}$

Step 2.7

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

$\frac{\sqrt{2 - 0 + 10 \cdot 0}}{7 - 5 \cdot 0}$

Step 2.8

Simplify the answer.

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

Multiply $- 1$ by $0$ .

$\frac{\sqrt{2 + 0 + 10 \cdot 0}}{7 - 5 \cdot 0}$

Step 2.8.1.2

Multiply $10$ by $0$ .

$\frac{\sqrt{2 + 0 + 0}}{7 - 5 \cdot 0}$

Step 2.8.1.3

Add $2 + 0$ and $0$ .

$\frac{\sqrt{2 + 0}}{7 - 5 \cdot 0}$

Step 2.8.1.4

Add $2$ and $0$ .

$\frac{\sqrt{2}}{7 - 5 \cdot 0}$

Step 2.8.2

Simplify the denominator.

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Step 2.8.2.1

Multiply $- 5$ by $0$ .

$\frac{\sqrt{2}}{7 + 0}$

Step 2.8.2.2

Add $7$ and $0$ .

$\frac{\sqrt{2}}{7}$

Step 3

Evaluate $lim_{x \to - \infty} \frac{\sqrt{2 x^{2} - x + 10}}{7 x - 5}$ 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 $- \sqrt{x^{2}} = x$ .

$lim_{x \to - \infty} \frac{- \sqrt{\frac{2 x^{2}}{x^{2}} + \frac{- x}{x^{2}} + \frac{10}{x^{2}}}}{\frac{7 x}{x} + \frac{- 5}{x}}$

Step 3.2

Evaluate the limit.

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Step 3.2.1

Simplify each term.

$lim_{x \to - \infty} \frac{- \sqrt{\frac{2 x^{2}}{x^{2}} + \frac{- x}{x^{2}} + \frac{10}{x^{2}}}}{7 - \frac{5}{x}}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{- \sqrt{\frac{2 x^{2}}{x^{2}} + \frac{- x}{x^{2}} + \frac{10}{x^{2}}}}{7 - \frac{5}{x}}$

Step 3.2.2.1.2

Divide $2$ by $1$ .

$lim_{x \to - \infty} \frac{- \sqrt{2 + \frac{- x}{x^{2}} + \frac{10}{x^{2}}}}{7 - \frac{5}{x}}$

Step 3.2.2.2

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

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Step 3.2.2.2.1

Factor $x$ out of $- x$ .

$lim_{x \to - \infty} \frac{- \sqrt{2 + \frac{x \cdot - 1}{x^{2}} + \frac{10}{x^{2}}}}{7 - \frac{5}{x}}$

Step 3.2.2.2.2

Cancel the common factors.

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Step 3.2.2.2.2.1

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

$lim_{x \to - \infty} \frac{- \sqrt{2 + \frac{x \cdot - 1}{x \cdot x} + \frac{10}{x^{2}}}}{7 - \frac{5}{x}}$

Step 3.2.2.2.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{- \sqrt{2 + \frac{x \cdot - 1}{x \cdot x} + \frac{10}{x^{2}}}}{7 - \frac{5}{x}}$

Step 3.2.2.2.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{- \sqrt{2 + \frac{- 1}{x} + \frac{10}{x^{2}}}}{7 - \frac{5}{x}}$

Step 3.2.2.3

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{- \sqrt{2 - \frac{1}{x} + \frac{10}{x^{2}}}}{7 - \frac{5}{x}}$

Step 3.2.3

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

$\frac{lim_{x \to - \infty} - \sqrt{2 - \frac{1}{x} + \frac{10}{x^{2}}}}{lim_{x \to - \infty} 7 - \frac{5}{x}}$

Step 3.2.4

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

$\frac{- lim_{x \to - \infty} \sqrt{2 - \frac{1}{x} + \frac{10}{x^{2}}}}{lim_{x \to - \infty} 7 - \frac{5}{x}}$

Step 3.2.5

Move the limit under the radical sign.

$\frac{- \sqrt{lim_{x \to - \infty} 2 - \frac{1}{x} + \frac{10}{x^{2}}}}{lim_{x \to - \infty} 7 - \frac{5}{x}}$

Step 3.2.6

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

$\frac{- \sqrt{lim_{x \to - \infty} 2 - lim_{x \to - \infty} \frac{1}{x} + lim_{x \to - \infty} \frac{10}{x^{2}}}}{lim_{x \to - \infty} 7 - \frac{5}{x}}$

Step 3.2.7

Evaluate the limit of $2$ which is constant as $x$ approaches $- \infty$ .

$\frac{- \sqrt{2 - lim_{x \to - \infty} \frac{1}{x} + lim_{x \to - \infty} \frac{10}{x^{2}}}}{lim_{x \to - \infty} 7 - \frac{5}{x}}$

Step 3.3

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

$\frac{- \sqrt{2 - 0 + lim_{x \to - \infty} \frac{10}{x^{2}}}}{lim_{x \to - \infty} 7 - \frac{5}{x}}$

Step 3.4

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

$\frac{- \sqrt{2 - 0 + 10 lim_{x \to - \infty} \frac{1}{x^{2}}}}{lim_{x \to - \infty} 7 - \frac{5}{x}}$

Step 3.5

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

$\frac{- \sqrt{2 - 0 + 10 \cdot 0}}{lim_{x \to - \infty} 7 - \frac{5}{x}}$

Step 3.6

Evaluate the limit.

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Step 3.6.1

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

$\frac{- \sqrt{2 - 0 + 10 \cdot 0}}{lim_{x \to - \infty} 7 - lim_{x \to - \infty} \frac{5}{x}}$

Step 3.6.2

Evaluate the limit of $7$ which is constant as $x$ approaches $- \infty$ .

$\frac{- \sqrt{2 - 0 + 10 \cdot 0}}{7 - lim_{x \to - \infty} \frac{5}{x}}$

Step 3.6.3

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

$\frac{- \sqrt{2 - 0 + 10 \cdot 0}}{7 - 5 lim_{x \to - \infty} \frac{1}{x}}$

Step 3.7

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

$\frac{- \sqrt{2 - 0 + 10 \cdot 0}}{7 - 5 \cdot 0}$

Step 3.8

Simplify the answer.

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Step 3.8.1

Simplify the numerator.

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Step 3.8.1.1

Multiply $- 1$ by $0$ .

$\frac{- \sqrt{2 + 0 + 10 \cdot 0}}{7 - 5 \cdot 0}$

Step 3.8.1.2

Multiply $10$ by $0$ .

$\frac{- \sqrt{2 + 0 + 0}}{7 - 5 \cdot 0}$

Step 3.8.1.3

Add $2 + 0$ and $0$ .

$\frac{- \sqrt{2 + 0}}{7 - 5 \cdot 0}$

Step 3.8.1.4

Add $2$ and $0$ .

$\frac{- \sqrt{2}}{7 - 5 \cdot 0}$

Step 3.8.2

Simplify the denominator.

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Step 3.8.2.1

Multiply $- 5$ by $0$ .

$\frac{- \sqrt{2}}{7 + 0}$

Step 3.8.2.2

Add $7$ and $0$ .

$\frac{- \sqrt{2}}{7}$

Step 3.8.3

Move the negative in front of the fraction.

$- \frac{\sqrt{2}}{7}$

Step 4

List the horizontal asymptotes:

$y = \frac{\sqrt{2}}{7}, - \frac{\sqrt{2}}{7}$

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: $x = \frac{5}{7}$

Horizontal Asymptotes: $y = \frac{\sqrt{2}}{7}, - \frac{\sqrt{2}}{7}$

Cannot Find Oblique Asymptotes

Step 7