Question

For what values of x, if any, does `f(x) = 1/((12x-8)sin(pi+(3pi)/x)` have vertical asymptotes?

Answer 1

The function has vertical asymptotes in the following points:

• `x = 2/3`
• All the points of the form `\frac{3}{2k-1}`
• All the points of the form `\frac{3}{2k}`

Note that, since `k` can be any integer number, the last two points can be summed up in

• All the points of the form `\frac{3}{n}`, with `n \in \mathbb{Z}`

Explanation:

A function usually has vertical asymptotes where it is not defined, either because of a zero denominator, or a zero argument in a logarithm.

In this case, we have to look for zeros of the denominator.

The denominator, in turn, is a multiplication of two factors, namely `(12x-8)` and `sin(\pi + \frac{3\pi}{x})`. A multiplication of two factor is zero if either of the two factors is zero, so let's examine them one by one.

`12x-8` equals zero if and only if...

`12x-8=0 \iff 12x = 8 \iff x = 8/12 = 2/3`

`sin(\pi + \frac{3\pi}{x})` equals zero if and only if the argument equals `2k\pi` or `\pi + 2k\pi`. Let's split the two cases again:

First case: the argument equals `2k\pi`

`\pi + \frac{3\pi}{x} = 2k\pi \iff \frac{3\pi}{x} = 2k\pi-\pi \iff \frac{x}{3\pi} = \frac{1}{2k\pi-\pi}`

`\iff x = \frac{3\pi}{2k\pi-\pi} = \frac{3}{2k-1}`

Second case: the argument equals `\pi+2k\pi`

`\pi + \frac{3\pi}{x} = \pi+2k\pi \iff \frac{3\pi}{x} = 2k\pi \iff \frac{x}{3\pi} = \frac{1}{2k\pi}`

`\iff x = \frac{3\pi}{2k\pi} = \frac{3}{2k}`