for the function f whose graph is given, state the following. (a) $lim_{x\rightarrowinfty}f(x)$ (b)…

Explanation:

Step1: Analyze limit as $x\to\infty$

As $x$ approaches positive - infinity, the graph of the function $y = f(x)$ approaches $y = 1$. So, $\lim_{x\to\infty}f(x)=1$.

Step2: Analyze limit as $x\to-\infty$

As $x$ approaches negative - infinity, the graph of the function $y = f(x)$ approaches $y = 1$. So, $\lim_{x\to-\infty}f(x)=1$.

Step3: Analyze limit as $x\to1$

As $x$ approaches $1$ from the left - hand side, $y = f(x)$ approaches $-\infty$, and as $x$ approaches $1$ from the right - hand side, $y = f(x)$ approaches $\infty$. So, $\lim_{x\to1}f(x)$ does not exist.

Step4: Analyze limit as $x\to3$

As $x$ approaches $3$ from the left - hand side and the right - hand side, $y = f(x)$ approaches $-\infty$. So, $\lim_{x\to3}f(x)=-\infty$.

Step5: Find vertical asymptotes

Vertical asymptotes occur where the function approaches $\pm\infty$. The function approaches $\pm\infty$ at $x = 1$ and $x = 3$. So the vertical asymptotes are $x = 1,x = 3$.

Step6: Find horizontal asymptotes

As $x\to\pm\infty$, the function approaches $y = 1$. So the horizontal asymptote is $y = 1$.

Answer:

(a) $1$ (b) $1$ (c) Does not exist (d) $-\infty$ (e) vertical: $x = 1,x = 3$; horizontal: $y = 1$