Let's analyze the graph and statements based on the image you uploaded.

1. Statement A: $\lim_{x \to -2} f(x) = -2$

• From the graph, as $x$ approaches $-2$, the function value approaches $-2$. Therefore, this statement is true.

2. Statement B: $f(x) \text{ is continuous at } x = 0$

• Continuity at $x = 0$ requires that:
  1. $f(0)$ is defined.
  2. $\lim_{x \to 0} f(x)$ exists.
  3. $\lim_{x \to 0} f(x) = f(0)$.
• Looking at the graph, there is a jump discontinuity at $x = 0$ (since the left and right limits don't match), so this statement is false.

3. Statement C: $\lim_{x \to 2} f(x) = \infty$

• At $x = 2$, the function approaches a finite value (which appears to be $2$), not $\infty$. Therefore, this statement is false.

4. Statement D: $\lim_{x \to 4} f(x) = f(4)$

• For $x = 4$, the graph shows that the limit as $x \to 4$ from the left is equal to $f(4)$, which is defined and equals $4$. So, this statement is true.

5. Statement E: $\lim_{x \to 0^+} f(x) = \lim_{x \to 0^-} f(x)$

• As $x$ approaches $0$ from the left ($x \to 0^-$), the function value approaches 2. As $x$ approaches 0 from the right ($x \to 0^+$), the function value approaches 4. These limits are not equal, so this statement is false.

Correct Answers:

• A and D are correct.

Would you like further details on any of these points?

5 Relative Questions:

1. What conditions must be met for a function to be continuous at a point?
2. How can we evaluate limits graphically at specific points?
3. What causes a discontinuity at $x = 0$ on the graph?
4. Why is $\lim_{x \to 2} f(x)$ not infinite in this case?
5. How can we determine if left-hand and right-hand limits are equal?

Tip:

Always check both the left-hand and right-hand limits when determining whether a function is continuous at a point or whether the limit exists.