3. (15 points) let $f(x)=\frac{4x - 4}{2x - 8}$. when finding a limit, write dne if any does not exist.\na)…

3. (15 points) let $f(x)=\frac{4x - 4}{2x - 8}$. when finding a limit, write dne if any does not exist.\na) find $lim_{x\rightarrowinfty}f(x)$ and $lim_{x\rightarrow-infty}f(x)$

3. (15 points) let $f(x)=\frac{4x - 4}{2x - 8}$. when finding a limit, write dne if any does not exist.\na) find $lim_{x\rightarrowinfty}f(x)$ and $lim_{x\rightarrow-infty}f(x)$

Answer

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 $\lim_{x\rightarrow\infty}f(x)$

As $x\rightarrow\infty$, $\frac{4}{x}\rightarrow0$ and $\frac{8}{x}\rightarrow0$. So $\lim_{x\rightarrow\infty}\frac{4-\frac{4}{x}}{2-\frac{8}{x}}=\frac{4 - 0}{2-0}=2$

Step3: Find $\lim_{x\rightarrow-\infty}f(x)$

As $x\rightarrow-\infty$, $\frac{4}{x}\rightarrow0$ and $\frac{8}{x}\rightarrow0$. So $\lim_{x\rightarrow-\infty}\frac{4-\frac{4}{x}}{2-\frac{8}{x}}=\frac{4 - 0}{2-0}=2$

Answer:

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