find the horizontal and vertical asymptotes for the function.\n\nf(x) = \\frac{2x - 4}{x^{2}-4}\n\nvertical…

Explanation:

Step1: Factor the function

First, factor the numerator and denominator. $f(x)=\frac{2x - 4}{x^{2}-4}=\frac{2(x - 2)}{(x + 2)(x - 2)}$. After canceling out the common - factor $(x - 2)$ (for $x\neq2$), we get $f(x)=\frac{2}{x + 2},x\neq2$.

Step2: Find the vertical asymptote

The vertical asymptote occurs where the denominator of the simplified function is zero. Set $x+2 = 0$, then $x=-2$.

Step3: Find the horizontal asymptote

For a rational function $y=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots+b_0}$, when $n\lt m$, the horizontal asymptote is $y = 0$. In the function $y=\frac{2}{x + 2}$, $n = 0$ (degree of the numerator) and $m = 1$ (degree of the denominator). So the horizontal asymptote is $y = 0$.

Answer:

Vertical asymptote: $x=-2$ Horizontal asymptote: $y = 0$