Trigonometry Examples

$\cos (\arctan (x))$

Step 1

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

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Step 3.1

Reduce.

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + x^{2}}$ as ${(1 + x^{2})}^{\frac{1}{2}}$ .

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

Step 3.1.2

Factor $1 + x^{2}$ out of ${(1 + x^{2})}^{\frac{1}{2}}$ .

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

Step 3.1.3

Cancel the common factors.

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Step 3.1.3.1

Multiply by $1$ .

$lim_{x \to \infty} \frac{(1 + x^{2}) {(1 + x^{2})}^{- \frac{1}{2}}}{(1 + x^{2}) \cdot 1}$

Step 3.1.3.2

Cancel the common factor.

$lim_{x \to \infty} \frac{(1 + x^{2}) {(1 + x^{2})}^{- \frac{1}{2}}}{(1 + x^{2}) \cdot 1}$

Step 3.1.3.3

Rewrite the expression.

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

Step 3.1.3.4

Divide ${(1 + x^{2})}^{- \frac{1}{2}}$ by $1$ .

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

Step 3.1.4

Move ${(1 + x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.2

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

$0$

Step 4

List the horizontal asymptotes:

$y = 0$

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.

No Vertical Asymptotes

Horizontal Asymptotes: $y = 0$

Cannot Find Oblique Asymptotes

Step 7