Question

1. (15 points) let $f(x)=\frac{4x - 4}{2x - 8}$. when finding a limit, write dne if any does not exist.

a) find $lim_{x
ightarrowinfty}f(x)$ and $lim_{x
ightarrow-infty}f(x)$

Explanation:

Step1: Simplify the function

Divide both numerator and denominator by x:
\[

$$\begin{align*} f(x)&=\frac{4x - 4}{2x-8}\\ &=\frac{4-\frac{4}{x}}{2-\frac{8}{x}} \end{align*}$$

\]

Step2: Find $\lim_{x

ightarrow\infty}f(x)$
As $x
ightarrow\infty$, $\frac{4}{x}
ightarrow0$ and $\frac{8}{x}
ightarrow0$.
So $\lim_{x
ightarrow\infty}\frac{4-\frac{4}{x}}{2-\frac{8}{x}}=\frac{4 - 0}{2-0}=2$

Step3: Find $\lim_{x

ightarrow-\infty}f(x)$
As $x
ightarrow-\infty$, $\frac{4}{x}
ightarrow0$ and $\frac{8}{x}
ightarrow0$.
So $\lim_{x
ightarrow-\infty}\frac{4-\frac{4}{x}}{2-\frac{8}{x}}=\frac{4 - 0}{2-0}=2$

Answer:

$\lim_{x
ightarrow\infty}f(x)=2$; $\lim_{x
ightarrow-\infty}f(x)=2$