Trigonometry Examples

$\csc (\arctan (x))$

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

$x = 0$

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

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Evaluate the limit.

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

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Cancel the common factor.

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

Rewrite the expression.

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

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

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

Move the limit under the radical sign.

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

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

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

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

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

Evaluate the limit.

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

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

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

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

Simplify the answer.

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

$\sqrt{0 + 1}$

Add $0$ and $1$ .

$\sqrt{1}$

Any root of $1$ is $1$ .

$1$

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

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Evaluate the limit.

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

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Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

Move the limit under the radical sign.

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

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

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

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

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

Evaluate the limit.

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

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

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

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

Simplify the answer.

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

$- \sqrt{0 + 1}$

Add $0$ and $1$ .

$- \sqrt{1}$

Any root of $1$ is $1$ .

$- 1 \cdot 1$

Multiply $- 1$ by $1$ .

$- 1$

List the horizontal asymptotes:

$y = 1, - 1$

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

Cannot Find Oblique Asymptotes

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 1, - 1$

Cannot Find Oblique Asymptotes