A vertical asymptote occurs when the function is divided by zero.
Thus, we have to evaluate the denominator.
In order for the denominator to be zero, one of the factors of the multiplication (in the denominator) must be zero.
So,
(a) x - 2 = 0
(b) x² - 1 = 0
(c) 2x - 3 = 0
(d) x + 2 =0
Let's isolate the x in the equations above to find the asymptotes.
(a) Adding 2 to both sides:
x - 2 + 2 = 0 +2
x = 2
(b) Adding 1 to both sides and then taking the square root:
x² - 1 + 1 = 0 + 1
x² = 1
√x² = ±√1
x² = ±1
(c) Adding 3 to both sides and then dividing the sides by 2:
2x - 3 + 3 = 0 + 3
2x = 3
2x/2 = 3/2
x = 3/2
(d) Subtracting 2 to both sides:
x + 2 - 2 = 0 - 2
x = -2
Answer: The vertical asymptotes are:
x = -2, x = -1, x = 1, x = 3/2, x = -2
