Question

Which of the following are vertical asymptotes of the function y = 2cot(3x) + 4? Check all that apply. A.x = pi/3 B.x = +/- pi/2 C.x = 2pi D.x = 0

Answer 1

Vertical asymptotes occur when the denominator of a rational is 0, whilst not zeroing out the numerator, making the rational, undefined, in this case

$\bf y=2cot(3x)+4\implies y=2\cdot \cfrac{cos(3x)}{sin(3x)}+4\impliedby \textit{if \underline{sin(3x)} turns to 0}\\\\ -------------------------------\\\\ \textit{let's check} \\\\\\ A)\qquad \cfrac{cos(3x)}{sin\left( 3\frac{\pi }{3} \right)}\implies \cfrac{cos(3x)}{sin\left( \pi \right)}\implies \cfrac{cos(3x)}{0}\impliedby unde f ined$

$\bf B)\qquad \cfrac{cos(3x)}{sin\left( 3\frac{\pm\pi }{2} \right)}\implies\cfrac{cos(3x)}{sin\left( \frac{\pm3\pi }{2} \right)}\implies \cfrac{cos(3x)}{\pm 1}\implies \pm cos(3x) \\\\\\ C)\qquad \cfrac{cos(3x)}{sin\left( 3(2\pi )\right)}\implies \cfrac{cos(3x)}{sin(6\pi )}\implies \cfrac{cos(3x)}{0}\impliedby unde f ined \\\\\\ D)\qquad \cfrac{cos(3x)}{sin(3(0))}\implies \cfrac{cos(3x)}{sin(0)}\implies \cfrac{cos(3x)}{0}\impliedby unde f ined$

Answer 2

Yeah it’s A C D for apex