9 mark for review let f be a function such that lim f(x) = ∞. which of the following statements must be…

Explanation:

Step1: Analyze the one - sided limit

Given $\lim_{x\rightarrow5^{-}}f(x)=\infty$. This is a left - hand limit as $x$ approaches 5 from the left side and the function values go to infinity.

Step2: Analyze option A

The left - hand limit $\lim_{x\rightarrow5^{-}}f(x)=\infty$ does not imply that the right - hand limit $\lim_{x\rightarrow5^{+}}f(x)=\infty$. The behavior of the function as $x$ approaches 5 from the right side is not given.

Step3: Analyze option B

The fact that $\lim_{x\rightarrow5^{-}}f(x)=\infty$ does not necessarily mean that $f(x)$ is undefined at $x = 5$. A function can have a limit behavior as $x$ approaches a value and still be defined at that value.

Step4: Analyze option C

If $\lim_{x\rightarrow5^{-}}f(x)=\infty$, then the graph of $y = f(x)$ has a vertical asymptote at $x = 5$. By the definition of a vertical asymptote, if either $\lim_{x\rightarrow a^{-}}f(x)=\pm\infty$ or $\lim_{x\rightarrow a^{+}}f(x)=\pm\infty$, then $x=a$ is a vertical asymptote of the graph of $y = f(x)$. Here $a = 5$.

Step5: Analyze option D

The limit is given as $x$ approaches 5 (not - 5), so $x=-5$ has no relation to the given limit $\lim_{x\rightarrow5^{-}}f(x)=\infty$.

Answer:

C. The graph of $f$ has a vertical asymptote at $x = 5$.