To analyze the function $f(x)$ based on the graph and answer which properties apply, let's examine each option step-by-step:

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

   • Looking at the graph near $x = -1$, the function appears to approach $+\infty$ as $x$ approaches $-1$ from the right. This suggests that this option is correct.

2. $\lim_{x \to 2} f(x) = 0$:

   • Observing the graph near $x = 2$, the function does not appear to approach 0. Instead, the value seems to be positive, so this option is incorrect.

3. $\lim_{x \to -\infty} f(x) = 0$:

   • As $x$ approaches $-\infty$, the function appears to approach the x-axis, which implies that the limit is indeed 0. This option is correct.

4. Has an asymptote at $x = -1$:

   • There is a vertical asymptote at $x = -1$, as the function approaches $+\infty$ and $-\infty$ on either side of $x = -1$. Therefore, this option is correct.

5. Has an asymptote $y = 0$ or $y = +\infty$:

   • The function appears to approach $y = 0$ as $x \to -\infty$, which implies a horizontal asymptote at $y = 0$. This option is correct.

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

   • On the left side of $x = -1$, the function approaches $-\infty$, which confirms this option is correct.

Summary of Correct Options:

• $\lim_{x \to -1^+} f(x) = +\infty$
• $\lim_{x \to -\infty} f(x) = 0$
• Has an asymptote at $x = -1$
• Has an asymptote $y = 0$ or $y = +\infty$
• $\lim_{x \to -1^-} f(x) = -\infty$

Would you like a detailed explanation of each limit behavior?

Related Questions:

1. What is the significance of horizontal asymptotes in analyzing limits as $x \to \infty$?
2. How can vertical asymptotes be identified based on the function’s equation?
3. What are the typical behaviors of a function around asymptotes?
4. How does the concept of one-sided limits help in understanding function behavior near discontinuities?
5. Can a function have multiple horizontal asymptotes?

Tip: When evaluating limits, consider both one-sided limits for a thorough understanding of function behavior near vertical asymptotes.