To solve the problem based on the image, let's analyze the graph of the function and then address each of the questions one by one.

Given:

The function is graphed, and we are asked to solve the following limits and continuity questions based on this graph.

a) $\lim_{x \to -4} f(x)$

To find the limit as $x$ approaches $-4$, we need to examine the graph at $x = -4$. Based on the graph:

• As $x \to -4^-$ (from the left), the graph approaches $y = 2$.
• As $x \to -4^+$ (from the right), the graph also approaches $y = 2$.

Since both sides approach the same value, we conclude: $\lim_{x \to -4} f(x) = 2$

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

For $x \to -2$:

• As $x \to -2^-$ (from the left), the graph approaches $y = 1$.
• As $x \to -2^+$ (from the right), the graph approaches $y = 1$.

Again, since both sides approach the same value: $\lim_{x \to -2} f(x) = 1$

c) $\lim_{x \to 4} f(x)$

At $x = 4$, we notice a discontinuity:

• As $x \to 4^-$ (from the left), the graph approaches $y = 1$.
• As $x \to 4^+$ (from the right), the graph approaches $y = 2$.

Since the left and right limits are not equal, the limit does not exist: $\lim_{x \to 4} f(x) \text{ does not exist}.$

d) Indique los intervalos donde la función es continua.

The function is continuous where there are no jumps, breaks, or holes. Based on the graph, the function is continuous in the following intervals:

• From $x = -4$ to $x = -2$ (excluding $x = -2$, which is a removable discontinuity).
• From $x = -2$ to $x = 0$ (including these points).
• From $x = 0$ to $x = 2$.
• From $x = 2$ to $x = 4$ (excluding $x = 4$, where there is a discontinuity).

Therefore, the intervals where the function is continuous are: $(-4, -2) \cup (-2, 0] \cup (0, 2) \cup (2, 4)$

e) Los valores de $x$ donde la función es discontinua. Además, indique el tipo de discontinuidad.

There are discontinuities at the following points:

• At $x = -2$: Removable discontinuity (since the limit exists but the function value does not match the limit).
• At $x = 4$: Jump discontinuity (since the left and right limits do not match).

So, the function has discontinuities at: $x = -2 \text{ (removable)}, \quad x = 4 \text{ (jump)}.$

Would you like more details or have any other questions?
Here are five related questions you might consider:

1. How do you identify the type of discontinuity from a graph?
2. What is the difference between a removable and jump discontinuity?
3. How can you determine limits for more complex piecewise functions?
4. How does a function's continuity relate to differentiability?
5. What are the conditions for a function to be continuous over a given interval?

Tip: Always check both left-hand and right-hand limits when evaluating limits at points of potential discontinuity.