Question

Which of the following are vertical asymptotes of the function y = 3cot(1/2x) - 4?

Answer 1

x=0 or ±2 nπ where n belongs to natural numbers.

Explanation:

The " vertical asymptote " is a vertical line that the graph of a function approaches but never touches. To find the vertical asymptotes of a rational function we set denominator=0.

We are given function y=3 cot((1/2)x)-4

which could also be written as
`y=3*(cos((1/2)x))/(sin((1/2)x)) -4\\\\y=(3 cos((1/2)x)-4 sin((1/2)x))/(sin((1/2)x))`

for denominator to be equal to 0 we must have sin((1/2)x)=0

⇒ (1/2)x=0

⇒ x=0 or ±2 nπ where n belongs to natural numbers.

Hence, the vertical asymptotes of the given function is x=0 or ±2 nπ where n belongs to natural numbers.

Answer 2

A vertical asymptote is a line that the graph of the function does not cross.
If:
`\lim_(x \to a) 3 cot (1/2 x)-4 =` +/- ∞
then the line x = a is a vertical asymptote.
For x = 0:
f ( 0 ) = 3 * cot 0 - 4 = ∞
For x = +/- 2 π :
f ( 2 π ) = 3 * cot π - 4 = - ∞
Answer:
A ) x = 0 and C ) x = +/- 2π