Answer:
Option a)
Step-by-step explanation:
To get the vertical asymptotes of the function f(x) you must find the limit when x tends k of f(x). If this limit tends to infinity then x = k is a vertical asymptote of the function.
[tex]\lim_{x\to\\2}\frac{x^3}{(x-2)^4} \\\\\\lim_{x\to\\2}\frac{2^3}{(2-2)^4}\\\\\lim_{x\to\\2}\frac{2^3}{(0)^4} = \infty[/tex]
Then. x = 2 it's a vertical asintota.
To obtain the horizontal asymptote of the function take the following limit:
[tex]\lim_{x \to \infty}\frac{x^3}{(x-2)^4}[/tex]
if [tex]\lim_{x \to \infty}\frac{x^3}{(x-2)^4} = b[/tex] then y = b is horizontal asymptote
Then:
[tex]\lim_{x \to \infty}\frac{x^3}{(x-2)^4} \\\\\\lim_{x \to \infty}\frac{1}{(\infty)} = 0[/tex]
Therefore y = 0 is a horizontal asymptote of f(x).
Then the correct answer is the option a) x = 2, y = 0
