Let's analyze the graph of $f(x)$ based on the given image to answer the questions.

(a) $f(-3)$

At $x = -3$, the graph shows a point at approximately $y = 6$.

• Answer: 6

(b) $f(2)$

At $x = 2$, the function is undefined since there is no closed dot on the graph at $x = 2$.

• Answer: DNE (Does Not Exist)

(c) $\lim_{x \to -3} f(x)$

The limit as $x \to -3$ from both sides approaches the value of the function at $x = -3$, which is 6.

• Answer: 6

(d) $\lim_{x \to -3^+} f(x)$

As $x$ approaches $-3$ from the right, the graph approaches the point $y = 6$.

• Answer: 6

(e) $\lim_{x \to -2} f(x)$

As $x$ approaches $-2$ from both sides, the function value approaches 2.

• Answer: 2

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

As $x$ approaches $-2$ from the right, the graph still approaches 2.

• Answer: 2

(g) $\lim_{x \to -2^-} f(x)$

As $x$ approaches $-2$ from the left, the limit is also 2.

• Answer: 2

(h) $\lim_{x \to -2} f(x)$ (repeated)

This is the same as (e).

• Answer: 2

(i) $\lim_{x \to 2^-} f(x)$

As $x$ approaches 2 from the left, the function approaches negative infinity.

• Answer: $-\infty$

(j) $\lim_{x \to 2^+} f(x)$

As $x$ approaches 2 from the right, the function approaches positive infinity.

• Answer: $+\infty$

(k) $\lim_{x \to -5} f(x)$

As $x$ approaches $-5$, the function approaches negative infinity.

• Answer: $-\infty$

(l) $\lim_{x \to 5^+} f(x)$

As $x$ approaches 5 from the right, the graph behaves erratically (oscillates), so the limit does not exist.

• Answer: DNE

(m) $\lim_{x \to 5} f(x)$

The function oscillates heavily near $x = 5$, indicating the limit does not exist.

• Answer: DNE

(n) $\lim_{x \to 7} f(x)$

As $x$ approaches $7$, the function value approaches $y = 2$.

• Answer: 2

(o) Vertical Asymptotes

Vertical asymptotes exist where the function tends to $\infty$ or $-\infty$. From the graph, there appears to be a vertical asymptote at $x = 2$.

• Answer: $x = 2$

Let me know if you'd like further clarification on any of these answers.

Follow-up questions:

1. What are the general conditions for a limit to exist?
2. How does the behavior of the function near $x = 5$ lead to a "does not exist" limit?
3. How do we identify vertical asymptotes on a graph?
4. Why does the limit approach infinity at $x = 2$?
5. What other types of discontinuities can a function have?

Tip: A limit exists only if the left-hand and right-hand limits are equal at a given point.