I see that you've uploaded an image containing several questions based on a graph. Let me extract the details and work through the parts one by one.

Part 1:

The question is asking about the limit of $f(x)$ as $x$ approaches $-1$ from the left-hand side (denoted as $x \to -1^-$).

From the graph, as $x$ approaches $-1$ from the left, the function $f(x)$ decreases without bound. Hence: $\lim_{x \to -1^-} f(x) = -\infty$

Part 2:

This part is asking about the limit as $x \to -1^+$ (approaching from the right side).

From the graph, as $x$ approaches $-1$ from the right, the function increases without bound. Therefore: $\lim_{x \to -1^+} f(x) = \infty$

Part 3:

Here, it asks for the limit of $f(x)$ as $x$ approaches infinity, $\lim_{x \to \infty} f(x)$.

From the graph, as $x$ goes to infinity, it looks like $f(x)$ approaches the horizontal asymptote at $y = 0$. So: $\lim_{x \to \infty} f(x) = 0$

Part 4:

This part asks for the limit as $x \to -\infty$.

From the graph, as $x$ goes to negative infinity, $f(x)$ again approaches the horizontal asymptote at $y = 0$. Thus: $\lim_{x \to -\infty} f(x) = 0$

Part 5:

This part asks if the graph is increasing over at least one interval and if it is decreasing over at least one interval.

From the graph:

• The function is increasing on the interval $(-1, \infty)$.
• The function is decreasing on the interval $(-\infty, -1)$.

So the answer to both questions is True.

Part 6:

This part asks for the interval over which the graph is increasing. From the graph, the function is increasing for: $(-1, \infty)$

Would you like a further breakdown of any specific part or the next steps?

Here are 5 related questions for practice:

1. What happens to $f(x)$ as $x \to 0^+$?
2. Identify the vertical asymptote(s) for the graph.
3. What would be the domain of the function $f(x)$?
4. Does the function ever cross the horizontal asymptote? Why or why not?
5. What is the range of the function based on the graph?

Tip:

When analyzing graphs for limits, always carefully examine behavior near key points (like where asymptotes occur) and at extreme values (as $x \to \infty$ or $x \to -\infty$).