Question

question
find all horizontal asymptotes of the following function.
f(x)=\frac{2 x+2}{x^{2}-x-2}
answer attempt 1 out of 2
one horizontal asymptote

Explanation:

Step1: Divide numerator and denominator by \(x^2\)

$$\lim_{x ightarrow\pm\infty}\frac{2x + 2}{x^2 - x - 2}=\lim_{x ightarrow\pm\infty}\frac{\frac{2x}{x^2}+\frac{2}{x^2}}{\frac{x^2}{x^2}-\frac{x}{x^2}-\frac{2}{x^2}}$$

Step2: Simplify the expression

$$=\lim_{x ightarrow\pm\infty}\frac{\frac{2}{x}+\frac{2}{x^2}}{1-\frac{1}{x}-\frac{2}{x^2}}$$

Step3: Evaluate the limit

As \(x
ightarrow\pm\infty\), \(\frac{2}{x}
ightarrow0\), \(\frac{2}{x^2}
ightarrow0\), \(\frac{1}{x}
ightarrow0\)
$$=\frac{0 + 0}{1-0 - 0}=0$$

Answer:

\(y = 0\)