Let's go through the limits one by one, based on the graph provided.

1. $\lim_{x \to -1^{-}} f(x)$:

   • This represents the left-hand limit as $x$ approaches $-1$ from the left. From the graph, as $x$ approaches $-1$ from the left, the function is oscillating but remains bounded around $0$. Thus: $\lim_{x \to -1^{-}} f(x) = 0$

2. $\lim_{x \to -1^{+}} f(x)$:

   • This represents the right-hand limit as $x$ approaches $-1$ from the right. From the graph, as $x$ approaches $-1$ from the right, the function diverges towards $-\infty$. Therefore: $\lim_{x \to -1^{+}} f(x) = -\infty$

3. $\lim_{x \to 1} f(x)$:

   • This is the two-sided limit as $x$ approaches $1$. As $x$ approaches $1$, the function seems to diverge towards $\infty$, implying that there is a vertical asymptote at $x = 1$. Thus: $\lim_{x \to 1} f(x) = \infty$

4. $\lim_{x \to \infty} f(x)$:

   • As $x$ approaches $\infty$, the function settles towards a horizontal asymptote. From the graph, the function appears to approach $0$ as $x$ increases. Hence: $\lim_{x \to \infty} f(x) = 0$

5. $\lim_{x \to -\infty} f(x)$:

   • As $x$ approaches $-\infty$, the function oscillates with a decaying amplitude, and it seems to approach $0$. Therefore: $\lim_{x \to -\infty} f(x) = 0$

Summary:

• $\lim_{x \to -1^{-}} f(x) = 0$
• $\lim_{x \to -1^{+}} f(x) = -\infty$
• $\lim_{x \to 1} f(x) = \infty$
• $\lim_{x \to \infty} f(x) = 0$
• $\lim_{x \to -\infty} f(x) = 0$

Would you like further details or explanations on any of these limits?

Here are five related questions:

1. What is the difference between a left-hand limit and a right-hand limit?
2. How can we identify a vertical asymptote from a function's graph?
3. What conditions lead to an infinite limit as $x$ approaches a certain value?
4. What is the significance of a horizontal asymptote in a function?
5. Can a function have multiple vertical asymptotes? Why or why not?

Tip: Always check the behavior of the function from both sides of a point when evaluating limits, especially near vertical asymptotes!