Determine the vertical asymptotes of the graph of the following functions
f(x)=cotx
f(x)=x-2/x^2-4
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.
To determine the vertical asymptotes of a function, we need to find the values of x for which the function approaches positive or negative infinity.
1. For the function f(x) = cot(x):
To find the vertical asymptotes for cot(x), we need to find the values of x where the function is undefined, i.e., where the denominator becomes zero.
The denominator of cot(x) is sin(x). The function cot(x) is undefined whenever sin(x) = 0.
So, sin(x) = 0 when x = kπ, where k is an integer. These are called the vertical asymptotes of cot(x).
Therefore, the vertical asymptotes for f(x) = cot(x) are x = kπ, where k is an integer.
2. For the function f(x) = (x-2)/(x^2-4):
To determine the vertical asymptotes of this function, we need to find the values of x when the denominator equals zero.
The denominator x^2 - 4 = 0 can be factored as (x - 2)(x + 2) = 0.
So, x = 2 and x = -2 are the vertical asymptotes of the function f(x) = (x-2)/(x^2-4).
In conclusion, the vertical asymptotes for f(x) = cot(x) are x = kπ, where k is an integer, and the vertical asymptotes for f(x) = (x-2)/(x^2-4) are x = 2 and x = -2.