The problem is about analyzing the behavior of a function $f(x)$ as $x$ approaches various values based on the given graph, focusing on limits. Here's a breakdown of each part based on the image:

• g. You are asked to select the correct choice for $\lim_{x \to -2} f(x)$.

  • Based on the graph, as $x \to -2$, $f(x)$ approaches positive infinity ($+\infty$), so the correct answer is A: $\lim_{x \to -2} f(x) = \infty$.

• h. You are asked to evaluate $\lim_{x \to 1^+} f(x)$, meaning the limit as $x$ approaches 1 from the right.

  • From the graph, as $x \to 1^+$, the function approaches $+\infty$, so the correct answer is $\infty$.

• i. You are asked to evaluate $\lim_{x \to 1^-} f(x)$, meaning the limit as $x$ approaches 1 from the left.

  • As $x \to 1^-$, the graph shows that $f(x)$ approaches $-\infty$, so the correct answer is $-\infty$.

• j. You are asked to evaluate $\lim_{x \to 1} f(x)$, but you are selecting between two choices.

  • Since the left-hand limit $\lim_{x \to 1^-} f(x) = -\infty$ and the right-hand limit $\lim_{x \to 1^+} f(x) = +\infty$, the two limits do not match. Therefore, the correct answer is B: The limit does not exist and is neither $-\infty$ nor $+\infty$.

• k. You are asked to evaluate $\lim_{x \to \infty} f(x)$.

  • As $x$ approaches infinity, $f(x)$ approaches 1, so the correct answer is $1$.

If you need more clarification or details on any part, feel free to ask!

Additional Related Questions:

1. What is the difference between left-hand and right-hand limits?
2. How does vertical asymptotic behavior affect the limit of a function?
3. Why does a limit not exist when the left and right limits differ?
4. Can a function have finite limits at infinity, and what does it signify graphically?
5. How do we determine horizontal asymptotes from the graph of a function?

Tip:

Always check both the left-hand and right-hand limits when dealing with points of discontinuity, as they provide insight into whether the limit exists.