f(x)=\frac{x^{2}+4}{4x^{2}-4x - 8}\nname the horizontal asymptote(s).\ny = 1/4\ncomplete\nbecause\n○ m <…

f(x)=\frac{x^{2}+4}{4x^{2}-4x - 8}\nname the horizontal asymptote(s).\ny = 1/4\ncomplete\nbecause\n○ m < n\n○ m = n\n○ a_m < b_n\n○ a_m = b_n\n○ this is where the function is undefined\ndone

f(x)=\frac{x^{2}+4}{4x^{2}-4x - 8}\nname the horizontal asymptote(s).\ny = 1/4\ncomplete\nbecause\n○ m < n\n○ m = n\n○ a_m < b_n\n○ a_m = b_n\n○ this is where the function is undefined\ndone

Answer

Explanation:

Step1: Identify degrees of polynomials

For a rational function $f(x)=\frac{a_mx^m+\cdots}{b_nx^n+\cdots}$, in $f(x)=\frac{x^2 + 4}{4x^2-4x - 8}$, the degree of the numerator $m = 2$ and the degree of the denominator $n=2$.

Step2: Recall horizontal - asymptote rule

When $m = n$ for a rational function $y=\frac{a_mx^m+\cdots}{b_nx^n+\cdots}$, the horizontal asymptote is $y=\frac{a_m}{b_n}$. Here $a_m = 1$ (the leading - coefficient of the numerator) and $b_n=4$ (the leading - coefficient of the denominator), so $y=\frac{1}{4}$.

Answer:

$m = n$