Which of the following are the equations of lines that are asymptotes of the graph of `y = (x^(2)-64)/((3x+4)(x-5))` ?
I. `x=-8`
II. X = 5
III. `y = (1)/(3)`

Options

A. I only
B. II only
C. I and II only
D. II and III only

Answer

D

Explanation

The denominator `(3x+4)(x-5)` is equal to 0 if `3x+4=0` or if `x-5=0`. If `3x+4=0`, then `3x=-4`, and `x=-(4)/(3)`. If `x-5=0`, then x = 5. The vertical asymptotes are `x =-(4)/(3)` and x = 5. Values of x with large absolute values can be large positive values of x and can also be negative values of x , both are far from 0 on the number line. For values of x with large absolute values, in `(x^(2)-64)/((3x+4)(x-5)), x^(2), x`, and 3x have much greater absolute values than `-64`, 4, and `-5`. So when x approaches values that have a large absolute value, `(x^(2)-64)/((3x+4)(x-5))` approaches `(x^(2))/((3x)(x))=(x^(2))/(3x^(2))=(1)/(3)`. So the horizontal asymptote is `y=(1)/(3)`. Thus, `x=-8` is not an asymptote, while x = 5 and `y = (1)/(3)` are asymptotes. (D) is correct.