Precalculus Examples

$y = \cot (\frac{x}{2})$

Step 1

For any $y = \cot (x)$ , vertical asymptotes occur at $x = n π$ , where $n$ is an integer. Use the basic period for $y = \cot (x)$ , $(0, π)$ , to find the vertical asymptotes for $y = \cot (\frac{x}{2})$ . Set the inside of the cotangent function, $b x + c$ , for $y = a \cot (b x + c) + d$ equal to $0$ to find where the vertical asymptote occurs for $y = \cot (\frac{x}{2})$ .

$\frac{x}{2} = 0$

Step 2

Set the numerator equal to zero.

$x = 0$

Step 3

Set the inside of the cotangent function $\frac{x}{2}$ equal to $π$ .

$\frac{x}{2} = π$

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 π$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 π$

Step 4.2.1.2

Rewrite the expression.

$x = 2 π$

Step 5

The basic period for $y = \cot (\frac{x}{2})$ will occur at $(0, 2 π)$ , where $0$ and $2 π$ are vertical asymptotes.

$(0, 2 π)$

Step 6

Find the period $\frac{π}{| b |}$ to find where the vertical asymptotes exist.

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Step 6.1

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{2}}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 2$

Step 6.3

Move $2$ to the left of $π$ .

$2 π$

Step 7

The vertical asymptotes for $y = \cot (\frac{x}{2})$ occur at $0$ , $2 π$ , and every $2 π n$ , where $n$ is an integer.

$x = 2 π n$

Step 8

Cotangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = 2 π n$ where $n$ is an integer

Step 9