Question

Determine the equations of the vertical and horizontal asymptotes, if any, for y=x^3/(x-2)^4

Answer 1

ANSWER

The correct answer is A.

EXPLANATION

The given function is,

$y = \frac{ {x}^{3} }{ {(x - 2)}^{4} }$

To find the vertical asymptote, we equate the denominator to zero.

This implies that,

${(x - 2)}^{4} = 0$

$x - 2 = 0$

$x = 2$

To find the horizontal asymptote,we take limit to infinity.

$lim_{x\rightarrow \infty} \frac{ {x}^{3} }{ {(x - 2)}^{4} } =0$

The horizontal asymptotes is

$y=0$

Answer 2

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.

$\lim_{x\to\\2}\frac{x^3}{(x-2)^4} \\\\\\lim_{x\to\\2}\frac{2^3}{(2-2)^4}\\\\\lim_{x\to\\2}\frac{2^3}{(0)^4} = \infty$

Then. x = 2 it's a vertical asintota.

To obtain the horizontal asymptote of the function take the following limit:

$\lim_{x \to \infty}\frac{x^3}{(x-2)^4}$

if $\lim_{x \to \infty}\frac{x^3}{(x-2)^4} = b$ then y = b is horizontal asymptote

Then:

$\lim_{x \to \infty}\frac{x^3}{(x-2)^4} \\\\\\lim_{x \to \infty}\frac{1}{(\infty)} = 0$

Therefore y = 0 is a horizontal asymptote of f(x).

Then the correct answer is the option a) x = 2, y = 0