state the vertical asymptote(s) and determine the end behavior of the rational function ( f(x)=\frac{-2}{x…

Explanation:

Step1: Find vertical asymptote

For a rational function (y = \frac{N(x)}{D(x)}), vertical asymptotes occur where (D(x)=0) (assuming (N(x)\neq0) at that point). For (f(x)=\frac{-2}{x - 5}), set (x-5 = 0). (x=5)

Step2: Determine end - behavior

As (x\to\pm\infty), we consider the limit of (f(x)=\frac{-2}{x - 5}). We know that (\lim_{x\to\pm\infty}\frac{-2}{x - 5}). Since (\lim_{x\to\pm\infty}\frac{1}{x}=0), for (y=\frac{-2}{x - 5}), we can rewrite it as (y=\frac{-2}{x(1-\frac{5}{x})}) (\lim_{x\to-\infty}\frac{-2}{x - 5}=0) (because as (x\to-\infty), (\frac{-2}{x-5}\approx\frac{-2}{x}) and (\lim_{x\to-\infty}\frac{-2}{x}=0)) (\lim_{x\to+\infty}\frac{-2}{x - 5}=0) (because as (x\to+\infty), (\frac{-2}{x-5}\approx\frac{-2}{x}) and (\lim_{x\to+\infty}\frac{-2}{x}=0))

Answer:

Equation(s) of vertical asymptote(s): (x = 5) As (x\to-\infty,f(x)\to0) As (x\to+\infty,f(x)\to0)