Trigonometry Examples

$\frac{1.6 x - \sqrt{x^{2} + 49}}{\sqrt{x^{2} + 49}}$

Step 1

Find where the expression $\frac{0.2 (8 \sqrt{x^{2} + 49} x - 5 x^{2} - 245)}{x^{2} + 49}$ is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

Evaluate $lim_{x \to \infty} \frac{0.2 (8 \sqrt{x^{2} + 49} x - 5 x^{2} - 245)}{x^{2} + 49}$ to find the horizontal asymptote.

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Step 3.1

Move the term $0.2$ outside of the limit because it is constant with respect to $x$ .

$0.2 lim_{x \to \infty} \frac{8 \sqrt{x^{2} + 49} x - 5 x^{2} - 245}{x^{2} + 49}$

Step 3.2

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

$0.2 lim_{x \to \infty} \frac{\frac{8 \sqrt{x^{2} + 49} x}{x^{2}} + \frac{- 5 x^{2}}{x^{2}} + \frac{- 245}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}$

Step 3.3

Evaluate the limit.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

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

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Step 3.3.1.1.1

Factor $x$ out of $8 \sqrt{x^{2} + 49} x$ .

$0.2 lim_{x \to \infty} \frac{\frac{x \cdot 8 \sqrt{x^{2} + 49}}{x^{2}} + \frac{- 5 x^{2}}{x^{2}} + \frac{- 245}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}$

Step 3.3.1.1.2

Cancel the common factors.

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Step 3.3.1.1.2.1

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

$0.2 lim_{x \to \infty} \frac{\frac{x \cdot 8 \sqrt{x^{2} + 49}}{x \cdot x} + \frac{- 5 x^{2}}{x^{2}} + \frac{- 245}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}$

Step 3.3.1.1.2.2

Cancel the common factor.

$0.2 lim_{x \to \infty} \frac{\frac{x \cdot 8 \sqrt{x^{2} + 49}}{x \cdot x} + \frac{- 5 x^{2}}{x^{2}} + \frac{- 245}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}$

Step 3.3.1.1.2.3

Rewrite the expression.

$0.2 lim_{x \to \infty} \frac{\frac{8 \sqrt{x^{2} + 49}}{x} + \frac{- 5 x^{2}}{x^{2}} + \frac{- 245}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Cancel the common factor.

$0.2 lim_{x \to \infty} \frac{\frac{8 \sqrt{x^{2} + 49}}{x} + \frac{- 5 x^{2}}{x^{2}} + \frac{- 245}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}$

Step 3.3.1.2.2

Divide $- 5$ by $1$ .

$0.2 lim_{x \to \infty} \frac{\frac{8 \sqrt{x^{2} + 49}}{x} - 5 + \frac{- 245}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}$

Step 3.3.1.3

Move the negative in front of the fraction.

$0.2 lim_{x \to \infty} \frac{\frac{8 \sqrt{x^{2} + 49}}{x} - 5 - \frac{245}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}$

Step 3.3.2

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

$0.2 lim_{x \to \infty} \frac{\frac{8 \sqrt{x^{2} + 49}}{x} - 5 - \frac{245}{x^{2}}}{1 + \frac{49}{x^{2}}}$

Step 3.3.3

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

$0.2 \frac{lim_{x \to \infty} \frac{8 \sqrt{x^{2} + 49}}{x} - 5 - \frac{245}{x^{2}}}{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}$

Step 3.3.4

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

$0.2 \frac{lim_{x \to \infty} \frac{8 \sqrt{x^{2} + 49}}{x} - lim_{x \to \infty} 5 - lim_{x \to \infty} \frac{245}{x^{2}}}{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}$

Step 3.3.5

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$0.2 \frac{8 lim_{x \to \infty} \frac{\sqrt{x^{2} + 49}}{x} - lim_{x \to \infty} 5 - lim_{x \to \infty} \frac{245}{x^{2}}}{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}$

Step 3.4

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

$0.2 \frac{8 lim_{x \to \infty} \frac{\sqrt{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}}{\frac{x}{x}} - lim_{x \to \infty} 5 - lim_{x \to \infty} \frac{245}{x^{2}}}{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}$

Step 3.5

Evaluate the limit.

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Step 3.5.1

Cancel the common factor of $x$ .

$0.2 \frac{8 lim_{x \to \infty} \frac{\sqrt{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}}{1} - lim_{x \to \infty} 5 - lim_{x \to \infty} \frac{245}{x^{2}}}{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}$

Step 3.5.2

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

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Step 3.5.2.1

Cancel the common factor.

$0.2 \frac{8 lim_{x \to \infty} \frac{\sqrt{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}}{1} - lim_{x \to \infty} 5 - lim_{x \to \infty} \frac{245}{x^{2}}}{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}$

Step 3.5.2.2

Rewrite the expression.

$0.2 \frac{8 lim_{x \to \infty} \frac{\sqrt{1 + \frac{49}{x^{2}}}}{1} - lim_{x \to \infty} 5 - lim_{x \to \infty} \frac{245}{x^{2}}}{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}$

Step 3.5.3

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

$0.2 \frac{8 \frac{lim_{x \to \infty} \sqrt{1 + \frac{49}{x^{2}}}}{lim_{x \to \infty} 1} - lim_{x \to \infty} 5 - lim_{x \to \infty} \frac{245}{x^{2}}}{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}$

Step 3.5.4

Move the limit under the radical sign.

$0.2 \frac{8 \frac{\sqrt{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}}{lim_{x \to \infty} 1} - lim_{x \to \infty} 5 - lim_{x \to \infty} \frac{245}{x^{2}}}{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}$

Step 3.5.5

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

$0.2 \frac{8 \frac{\sqrt{lim_{x \to \infty} 1 + lim_{x \to \infty} \frac{49}{x^{2}}}}{lim_{x \to \infty} 1} - lim_{x \to \infty} 5 - lim_{x \to \infty} \frac{245}{x^{2}}}{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}$

Step 3.5.6

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

$0.2 \frac{8 \frac{\sqrt{1 + lim_{x \to \infty} \frac{49}{x^{2}}}}{lim_{x \to \infty} 1} - lim_{x \to \infty} 5 - lim_{x \to \infty} \frac{245}{x^{2}}}{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}$

Step 3.5.7

Move the term $49$ outside of the limit because it is constant with respect to $x$ .

$0.2 \frac{8 \frac{\sqrt{1 + 49 lim_{x \to \infty} \frac{1}{x^{2}}}}{lim_{x \to \infty} 1} - lim_{x \to \infty} 5 - lim_{x \to \infty} \frac{245}{x^{2}}}{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}$

Step 3.6

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

$0.2 \frac{8 \frac{\sqrt{1 + 49 \cdot 0}}{lim_{x \to \infty} 1} - lim_{x \to \infty} 5 - lim_{x \to \infty} \frac{245}{x^{2}}}{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}$

Step 3.7

Evaluate the limit.

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Step 3.7.1

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

$0.2 \frac{8 \frac{\sqrt{1 + 49 \cdot 0}}{1} - lim_{x \to \infty} 5 - lim_{x \to \infty} \frac{245}{x^{2}}}{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}$

Step 3.7.2

Evaluate the limit of $5$ which is constant as $x$ approaches $\infty$ .

$0.2 \frac{8 \frac{\sqrt{1 + 49 \cdot 0}}{1} - 1 \cdot 5 - lim_{x \to \infty} \frac{245}{x^{2}}}{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}$

Step 3.7.3

Move the term $245$ outside of the limit because it is constant with respect to $x$ .

$0.2 \frac{8 \frac{\sqrt{1 + 49 \cdot 0}}{1} - 1 \cdot 5 - 245 lim_{x \to \infty} \frac{1}{x^{2}}}{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}$

Step 3.8

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

$0.2 \frac{8 \frac{\sqrt{1 + 49 \cdot 0}}{1} - 1 \cdot 5 - 245 \cdot 0}{lim_{x \to \infty} 1 + \frac{49}{x^{2}}}$

Step 3.9

Evaluate the limit.

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Step 3.9.1

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

$0.2 \frac{8 \frac{\sqrt{1 + 49 \cdot 0}}{1} - 1 \cdot 5 - 245 \cdot 0}{lim_{x \to \infty} 1 + lim_{x \to \infty} \frac{49}{x^{2}}}$

Step 3.9.2

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

$0.2 \frac{8 \frac{\sqrt{1 + 49 \cdot 0}}{1} - 1 \cdot 5 - 245 \cdot 0}{1 + lim_{x \to \infty} \frac{49}{x^{2}}}$

Step 3.9.3

Move the term $49$ outside of the limit because it is constant with respect to $x$ .

$0.2 \frac{8 \frac{\sqrt{1 + 49 \cdot 0}}{1} - 1 \cdot 5 - 245 \cdot 0}{1 + 49 lim_{x \to \infty} \frac{1}{x^{2}}}$

Step 3.10

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

$0.2 \frac{8 \frac{\sqrt{1 + 49 \cdot 0}}{1} - 1 \cdot 5 - 245 \cdot 0}{1 + 49 \cdot 0}$

Step 3.11

Simplify the answer.

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Step 3.11.1

Divide $\sqrt{1 + 49 \cdot 0}$ by $1$ .

$0.2 \frac{8 \sqrt{1 + 49 \cdot 0} - 1 \cdot 5 - 245 \cdot 0}{1 + 49 \cdot 0}$

Step 3.11.2

Simplify the numerator.

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Step 3.11.2.1

Multiply $49$ by $0$ .

$0.2 \frac{8 \sqrt{1 + 0} - 1 \cdot 5 - 245 \cdot 0}{1 + 49 \cdot 0}$

Step 3.11.2.2

Add $1$ and $0$ .

$0.2 \frac{8 \sqrt{1} - 1 \cdot 5 - 245 \cdot 0}{1 + 49 \cdot 0}$

Step 3.11.2.3

Any root of $1$ is $1$ .

$0.2 \frac{8 \cdot 1 - 1 \cdot 5 - 245 \cdot 0}{1 + 49 \cdot 0}$

Step 3.11.2.4

Multiply $8$ by $1$ .

$0.2 \frac{8 - 1 \cdot 5 - 245 \cdot 0}{1 + 49 \cdot 0}$

Step 3.11.2.5

Multiply $- 1$ by $5$ .

$0.2 \frac{8 - 5 - 245 \cdot 0}{1 + 49 \cdot 0}$

Step 3.11.2.6

Multiply $- 245$ by $0$ .

$0.2 \frac{8 - 5 + 0}{1 + 49 \cdot 0}$

Step 3.11.2.7

Add $8 - 5$ and $0$ .

$0.2 \frac{8 - 5}{1 + 49 \cdot 0}$

Step 3.11.2.8

Subtract $5$ from $8$ .

$0.2 \frac{3}{1 + 49 \cdot 0}$

Step 3.11.3

Simplify the denominator.

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Step 3.11.3.1

Multiply $49$ by $0$ .

$0.2 \frac{3}{1 + 0}$

Step 3.11.3.2

Add $1$ and $0$ .

$0.2 (\frac{3}{1})$

Step 3.11.4

Cancel the common factor of $0.2$ .

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Step 3.11.4.1

Factor $0.2$ out of $1$ .

$0.2 \frac{3}{0.2 \cdot 5}$

Step 3.11.4.2

Cancel the common factor.

$0.2 \frac{3}{0.2 \cdot 5}$

Step 3.11.4.3

Rewrite the expression.

$\frac{3}{5}$

Step 4

Evaluate $lim_{x \to - \infty} \frac{0.2 (8 \sqrt{x^{2} + 49} x - 5 x^{2} - 245)}{x^{2} + 49}$ to find the horizontal asymptote.

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Step 4.1

Move the term $0.2$ outside of the limit because it is constant with respect to $x$ .

$0.2 lim_{x \to - \infty} \frac{8 \sqrt{x^{2} + 49} x - 5 x^{2} - 245}{x^{2} + 49}$

Step 4.2

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

$0.2 lim_{x \to - \infty} \frac{\frac{8 \sqrt{x^{2} + 49} x}{x^{2}} + \frac{- 5 x^{2}}{x^{2}} + \frac{- 245}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}$

Step 4.3

Evaluate the limit.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

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

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Step 4.3.1.1.1

Factor $x$ out of $8 \sqrt{x^{2} + 49} x$ .

$0.2 lim_{x \to - \infty} \frac{\frac{x \cdot 8 \sqrt{x^{2} + 49}}{x^{2}} + \frac{- 5 x^{2}}{x^{2}} + \frac{- 245}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}$

Step 4.3.1.1.2

Cancel the common factors.

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Step 4.3.1.1.2.1

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

$0.2 lim_{x \to - \infty} \frac{\frac{x \cdot 8 \sqrt{x^{2} + 49}}{x \cdot x} + \frac{- 5 x^{2}}{x^{2}} + \frac{- 245}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}$

Step 4.3.1.1.2.2

Cancel the common factor.

$0.2 lim_{x \to - \infty} \frac{\frac{x \cdot 8 \sqrt{x^{2} + 49}}{x \cdot x} + \frac{- 5 x^{2}}{x^{2}} + \frac{- 245}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}$

Step 4.3.1.1.2.3

Rewrite the expression.

$0.2 lim_{x \to - \infty} \frac{\frac{8 \sqrt{x^{2} + 49}}{x} + \frac{- 5 x^{2}}{x^{2}} + \frac{- 245}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}$

Step 4.3.1.2

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

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Step 4.3.1.2.1

Cancel the common factor.

$0.2 lim_{x \to - \infty} \frac{\frac{8 \sqrt{x^{2} + 49}}{x} + \frac{- 5 x^{2}}{x^{2}} + \frac{- 245}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}$

Step 4.3.1.2.2

Divide $- 5$ by $1$ .

$0.2 lim_{x \to - \infty} \frac{\frac{8 \sqrt{x^{2} + 49}}{x} - 5 + \frac{- 245}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}$

Step 4.3.1.3

Move the negative in front of the fraction.

$0.2 lim_{x \to - \infty} \frac{\frac{8 \sqrt{x^{2} + 49}}{x} - 5 - \frac{245}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}$

Step 4.3.2

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

$0.2 lim_{x \to - \infty} \frac{\frac{8 \sqrt{x^{2} + 49}}{x} - 5 - \frac{245}{x^{2}}}{1 + \frac{49}{x^{2}}}$

Step 4.3.3

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

$0.2 \frac{lim_{x \to - \infty} \frac{8 \sqrt{x^{2} + 49}}{x} - 5 - \frac{245}{x^{2}}}{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}$

Step 4.3.4

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

$0.2 \frac{lim_{x \to - \infty} \frac{8 \sqrt{x^{2} + 49}}{x} - lim_{x \to - \infty} 5 - lim_{x \to - \infty} \frac{245}{x^{2}}}{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}$

Step 4.3.5

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$0.2 \frac{8 lim_{x \to - \infty} \frac{\sqrt{x^{2} + 49}}{x} - lim_{x \to - \infty} 5 - lim_{x \to - \infty} \frac{245}{x^{2}}}{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}$

Step 4.4

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

$0.2 \frac{8 lim_{x \to - \infty} \frac{- \sqrt{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}}{\frac{x}{x}} - lim_{x \to - \infty} 5 - lim_{x \to - \infty} \frac{245}{x^{2}}}{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}$

Step 4.5

Evaluate the limit.

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Step 4.5.1

Cancel the common factor of $x$ .

$0.2 \frac{8 lim_{x \to - \infty} \frac{- \sqrt{\frac{x^{2}}{x^{2}} + \frac{49}{x^{2}}}}{1} - lim_{x \to - \infty} 5 - lim_{x \to - \infty} \frac{245}{x^{2}}}{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}$

Step 4.5.2

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

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Step 4.5.2.1

Cancel the common factor.

Step 4.5.2.2

Rewrite the expression.

$0.2 \frac{8 lim_{x \to - \infty} \frac{- \sqrt{1 + \frac{49}{x^{2}}}}{1} - lim_{x \to - \infty} 5 - lim_{x \to - \infty} \frac{245}{x^{2}}}{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}$

Step 4.5.3

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

$0.2 \frac{8 \frac{lim_{x \to - \infty} - \sqrt{1 + \frac{49}{x^{2}}}}{lim_{x \to - \infty} 1} - lim_{x \to - \infty} 5 - lim_{x \to - \infty} \frac{245}{x^{2}}}{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}$

Step 4.5.4

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

$0.2 \frac{8 \frac{- lim_{x \to - \infty} \sqrt{1 + \frac{49}{x^{2}}}}{lim_{x \to - \infty} 1} - lim_{x \to - \infty} 5 - lim_{x \to - \infty} \frac{245}{x^{2}}}{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}$

Step 4.5.5

Move the limit under the radical sign.

$0.2 \frac{8 \frac{- \sqrt{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}}{lim_{x \to - \infty} 1} - lim_{x \to - \infty} 5 - lim_{x \to - \infty} \frac{245}{x^{2}}}{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}$

Step 4.5.6

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

$0.2 \frac{8 \frac{- \sqrt{lim_{x \to - \infty} 1 + lim_{x \to - \infty} \frac{49}{x^{2}}}}{lim_{x \to - \infty} 1} - lim_{x \to - \infty} 5 - lim_{x \to - \infty} \frac{245}{x^{2}}}{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}$

Step 4.5.7

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

$0.2 \frac{8 \frac{- \sqrt{1 + lim_{x \to - \infty} \frac{49}{x^{2}}}}{lim_{x \to - \infty} 1} - lim_{x \to - \infty} 5 - lim_{x \to - \infty} \frac{245}{x^{2}}}{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}$

Step 4.5.8

Move the term $49$ outside of the limit because it is constant with respect to $x$ .

$0.2 \frac{8 \frac{- \sqrt{1 + 49 lim_{x \to - \infty} \frac{1}{x^{2}}}}{lim_{x \to - \infty} 1} - lim_{x \to - \infty} 5 - lim_{x \to - \infty} \frac{245}{x^{2}}}{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}$

Step 4.6

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

$0.2 \frac{8 \frac{- \sqrt{1 + 49 \cdot 0}}{lim_{x \to - \infty} 1} - lim_{x \to - \infty} 5 - lim_{x \to - \infty} \frac{245}{x^{2}}}{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}$

Step 4.7

Evaluate the limit.

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Step 4.7.1

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

$0.2 \frac{8 \frac{- \sqrt{1 + 49 \cdot 0}}{1} - lim_{x \to - \infty} 5 - lim_{x \to - \infty} \frac{245}{x^{2}}}{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}$

Step 4.7.2

Evaluate the limit of $5$ which is constant as $x$ approaches $- \infty$ .

$0.2 \frac{8 \frac{- \sqrt{1 + 49 \cdot 0}}{1} - 1 \cdot 5 - lim_{x \to - \infty} \frac{245}{x^{2}}}{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}$

Step 4.7.3

Move the term $245$ outside of the limit because it is constant with respect to $x$ .

$0.2 \frac{8 \frac{- \sqrt{1 + 49 \cdot 0}}{1} - 1 \cdot 5 - 245 lim_{x \to - \infty} \frac{1}{x^{2}}}{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}$

Step 4.8

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

$0.2 \frac{8 \frac{- \sqrt{1 + 49 \cdot 0}}{1} - 1 \cdot 5 - 245 \cdot 0}{lim_{x \to - \infty} 1 + \frac{49}{x^{2}}}$

Step 4.9

Evaluate the limit.

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Step 4.9.1

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

$0.2 \frac{8 \frac{- \sqrt{1 + 49 \cdot 0}}{1} - 1 \cdot 5 - 245 \cdot 0}{lim_{x \to - \infty} 1 + lim_{x \to - \infty} \frac{49}{x^{2}}}$

Step 4.9.2

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

$0.2 \frac{8 \frac{- \sqrt{1 + 49 \cdot 0}}{1} - 1 \cdot 5 - 245 \cdot 0}{1 + lim_{x \to - \infty} \frac{49}{x^{2}}}$

Step 4.9.3

Move the term $49$ outside of the limit because it is constant with respect to $x$ .

$0.2 \frac{8 \frac{- \sqrt{1 + 49 \cdot 0}}{1} - 1 \cdot 5 - 245 \cdot 0}{1 + 49 lim_{x \to - \infty} \frac{1}{x^{2}}}$

Step 4.10

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

$0.2 \frac{8 \frac{- \sqrt{1 + 49 \cdot 0}}{1} - 1 \cdot 5 - 245 \cdot 0}{1 + 49 \cdot 0}$

Step 4.11

Simplify the answer.

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Step 4.11.1

Divide $- \sqrt{1 + 49 \cdot 0}$ by $1$ .

$0.2 \frac{8 (- \sqrt{1 + 49 \cdot 0}) - 1 \cdot 5 - 245 \cdot 0}{1 + 49 \cdot 0}$

Step 4.11.2

Simplify the numerator.

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Step 4.11.2.1

Multiply $49$ by $0$ .

$0.2 \frac{8 (- \sqrt{1 + 0}) - 1 \cdot 5 - 245 \cdot 0}{1 + 49 \cdot 0}$

Step 4.11.2.2

Add $1$ and $0$ .

$0.2 \frac{8 (- \sqrt{1}) - 1 \cdot 5 - 245 \cdot 0}{1 + 49 \cdot 0}$

Step 4.11.2.3

Any root of $1$ is $1$ .

$0.2 \frac{8 (- 1 \cdot 1) - 1 \cdot 5 - 245 \cdot 0}{1 + 49 \cdot 0}$

Step 4.11.2.4

Multiply $8 (- 1 \cdot 1)$ .

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Step 4.11.2.4.1

Multiply $- 1$ by $1$ .

$0.2 \frac{8 \cdot - 1 - 1 \cdot 5 - 245 \cdot 0}{1 + 49 \cdot 0}$

Step 4.11.2.4.2

Multiply $8$ by $- 1$ .

$0.2 \frac{- 8 - 1 \cdot 5 - 245 \cdot 0}{1 + 49 \cdot 0}$

Step 4.11.2.5

Multiply $- 1$ by $5$ .

$0.2 \frac{- 8 - 5 - 245 \cdot 0}{1 + 49 \cdot 0}$

Step 4.11.2.6

Multiply $- 245$ by $0$ .

$0.2 \frac{- 8 - 5 + 0}{1 + 49 \cdot 0}$

Step 4.11.2.7

Add $- 8 - 5$ and $0$ .

$0.2 \frac{- 8 - 5}{1 + 49 \cdot 0}$

Step 4.11.2.8

Subtract $5$ from $- 8$ .

$0.2 \frac{- 13}{1 + 49 \cdot 0}$

Step 4.11.3

Simplify the denominator.

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Step 4.11.3.1

Multiply $49$ by $0$ .

$0.2 \frac{- 13}{1 + 0}$

Step 4.11.3.2

Add $1$ and $0$ .

$0.2 (\frac{- 13}{1})$

Step 4.11.4

Cancel the common factor of $0.2$ .

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Step 4.11.4.1

Factor $0.2$ out of $1$ .

$0.2 \frac{- 13}{0.2 \cdot 5}$

Step 4.11.4.2

Cancel the common factor.

$0.2 \frac{- 13}{0.2 \cdot 5}$

Step 4.11.4.3

Rewrite the expression.

$\frac{- 13}{5}$

Step 4.11.5

Move the negative in front of the fraction.

$- \frac{13}{5}$

Step 5

List the horizontal asymptotes:

$y = \frac{3}{5}, - \frac{13}{5}$

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: $y = \frac{3}{5}, - \frac{13}{5}$

Cannot Find Oblique Asymptotes

Step 8