Let's analyze the questions in the image one by one.

The image asks for evaluations of specific limits based on the graph of a function $f(x)$.

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

To evaluate this limit, observe the behavior of $f(x)$ as $x$ approaches $4$ from both sides on the graph:

• The limit value appears to be around $-1$, since as $x$ approaches $4$, the $y$-values of $f(x)$ are getting closer to $-1$.

Thus, the answer is: $\lim_{{x \to 4}} f(x) = -1$

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

This asks for the right-hand limit of $f(x)$ as $x$ approaches $-3$ from the positive (right) side.

• Looking at the graph, as $x$ approaches $-3$ from the right, the $y$-values approach $-5$.

So the answer is: $\lim_{{x \to -3^+}} f(x) = -5$

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

This asks for the left-hand limit of $f(x)$ as $x$ approaches $-3$ from the negative (left) side.

• Observing the graph, as $x$ approaches $-3$ from the left, the $y$-values appear to approach $3$.

Therefore, the answer is: $\lim_{{x \to -3^-}} f(x) = 3$

Would you like further details or explanations on these limit evaluations?

Related Questions:

1. How do we determine if a limit exists at a particular point?
2. What is the difference between a left-hand limit and a right-hand limit?
3. How do we find the limit of a function at a discontinuity?
4. What methods can be used to estimate limits when no graph is provided?
5. Why is the limit notation important in calculus?

Tip:

When evaluating limits from a graph, always check both sides of the point to see if they approach the same value. If not, the limit at that point does not exist.