Let's go through the questions using the given graph of the function $f$:

(a) Finding Limits

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

   • As $x$ approaches 2 from the left, $f(x)$ approaches a vertical asymptote, heading towards $+\infty$.

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

   • As $x$ approaches 2 from the right, $f(x)$ also heads towards a vertical asymptote, but this time going towards $-\infty$.

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

   • As $x$ approaches 3, the function seems to approach a finite value. By observing the graph, the limit appears to be $1$.

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

   • As $x$ approaches 4 from the left, $f(x)$ approaches the value at the open circle, which seems to be at $y = 2$.

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

   • As $x$ approaches 0, $f(x)$ approaches $0$ (since $f(0) = 0$).

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

   • As $x$ approaches $-3$, the function seems to head towards a vertical asymptote at $y \to -\infty$.

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

   • As $x$ goes to infinity, the function levels off towards a horizontal asymptote, approaching $y = 2$.

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

   • As $x$ goes to negative infinity, the function levels off towards a horizontal asymptote at $y = 0$.

(b) Horizontal Asymptotes

• From the graph:
  1. $y = 0$ as $x \to -\infty$
  2. $y = 2$ as $x \to \infty$

(c) Vertical Asymptotes

• The function has vertical asymptotes at:
  1. $x = -3$
  2. $x = 2$

(d) Discontinuities

• The function is discontinuous at the following points:
  1. $x = -3$ (vertical asymptote)
  2. $x = 2$ (vertical asymptote)
  3. $x = 4$ (jump discontinuity where the function has a hole at $x = 4$ and a different point value elsewhere)

Would you like more detailed explanations for any of these limits or concepts?

Relative Questions

1. What are the types of discontinuities in a function and how are they identified?
2. How do you determine the behavior of a function as it approaches a vertical asymptote?
3. How are horizontal asymptotes different from slant asymptotes, and when do slant asymptotes occur?
4. How can the limit at a point help in determining the continuity of a function?
5. What role do asymptotes play in understanding the end behavior of a function?

Tip

When analyzing a graph for limits, focus on the direction from which $x$ approaches a point, as limits can differ from the left-hand side and right-hand side.