Determine the vertical asymptotes of the graph of the following functions

Question

f(x)=cotx
f(x)=x-2/x^2-4

Answer 1

To determine the vertical asymptotes of a function, we look for values of x that result in the function approaching infinity or negative infinity.

1. f(x) = cot(x):

The function cot(x) is equal to 1/tan(x). A vertical asymptote occurs when the tangent function is equal to zero. Tangent is equal to zero at x = (n * π), where n is an integer. Therefore, the vertical asymptotes for f(x) = cot(x) are x = nπ, where n is an integer.

2. f(x) = (x - 2)/(x^2 - 4):

To find vertical asymptotes for this function, we consider the values of x that make the denominator equal to zero. In this case, the denominator is x^2 - 4, which factors to (x - 2)(x + 2). Therefore, the vertical asymptotes occur when x - 2 = 0 or x + 2 = 0. Solving for x, we find that the vertical asymptotes are x = 2 and x = -2.