Question

question
find all horizontal asymptotes of the following function.
f(x)=\frac{5 x^{2}-42 x + 49}{10 x^{2}-14 x}
answer attempt 1 out of 2
one horizontal asymptote
no horizontal asymptotes
one horizontal asymptote
two horizontal asymptotes

Explanation:

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

$$\begin{align*} \lim_{x ightarrow\pm\infty}f(x)&=\lim_{x ightarrow\pm\infty}\frac{5x^{2}-42x + 49}{10x^{2}-14x}\\ &=\lim_{x ightarrow\pm\infty}\frac{5-\frac{42}{x}+\frac{49}{x^{2}}}{10-\frac{14}{x}} \end{align*}$$

Step2: Evaluate the limit

As \(x
ightarrow\pm\infty\), \(\frac{1}{x}
ightarrow0\) and \(\frac{1}{x^{2}}
ightarrow0\).
So \(\lim_{x
ightarrow\pm\infty}\frac{5-\frac{42}{x}+\frac{49}{x^{2}}}{10-\frac{14}{x}}=\frac{5 - 0+0}{10-0}=\frac{1}{2}\)

Answer:

One Horizontal Asymptote \(y = \frac{1}{2}\)