17. the graph of a function f is shown above. if lim f(x) exists and f is not continuous at b, then b = (a)…

Explanation:

Step1: Recall continuity and limit - existence conditions

A function $y = f(x)$ is continuous at $x = a$ if $\lim_{x\rightarrow a}f(x)=f(a)$, and $\lim_{x\rightarrow a}f(x)$ exists if $\lim_{x\rightarrow a^{-}}f(x)=\lim_{x\rightarrow a^{+}}f(x)$.

Step2: Analyze $x = 0$

At $x = 0$, the left - hand limit $\lim_{x\rightarrow0^{-}}f(x)=-\infty$ and the right - hand limit $\lim_{x\rightarrow0^{+}}f(x)$ is a finite value. So $\lim_{x\rightarrow0}f(x)$ does not exist.

Step3: Analyze $x = 1$

At $x = 1$, the function has a sharp turn. But $\lim_{x\rightarrow1^{-}}f(x)=\lim_{x\rightarrow1^{+}}f(x)$, so $\lim_{x\rightarrow1}f(x)$ exists. Also, $f(1)$ is defined and $\lim_{x\rightarrow1}f(x)=f(1)$, so the function is continuous at $x = 1$.

Step4: Analyze $x = 2$

At $x = 2$, the left - hand limit $\lim_{x\rightarrow2^{-}}f(x)$ and the right - hand limit $\lim_{x\rightarrow2^{+}}f(x)$ are not equal. So $\lim_{x\rightarrow2}f(x)$ does not exist.

Step5: Analyze $x = 3$

At $x = 3$, $\lim_{x\rightarrow3^{-}}f(x)$ and $\lim_{x\rightarrow3^{+}}f(x)$ are not equal. So $\lim_{x\rightarrow3}f(x)$ does not exist.

Answer:

A. 1