Let's analyze the graph and answer the given questions.

Observations from the Graph of $f(x)$:

• The function $f(x)$ appears to be a V-shaped graph, suggesting that it may represent an absolute value function, such as $f(x) = |x|$.
• There are sharp changes at $x = 1$ and possibly at other points.

Answering the Questions:

(a) Does $\lim_{x \to 3} f(x)$ exist?

• Yes, the graph approaches the same value from both the left and the right as $x \to 3$, indicating that the limit exists.

(b) Is $f(x)$ continuous at $x = 3$?

• Yes, the graph is connected and smooth at $x = 3$, indicating the function is continuous at that point.

(c) Is $f(x)$ differentiable at $x = 3$?

• Yes, there is no sharp corner or cusp at $x = 3$, meaning the function is differentiable there.

(d) Does $\lim_{x \to 1} f(x)$ exist?

• Yes, the limit exists because the graph approaches the same value from both sides as $x \to 1$.

(e) Is $f(x)$ continuous at $x = 1$?

• Yes, the function is continuous at $x = 1$, since there is no break or jump in the graph.

(f) Is $f(x)$ differentiable at $x = 1$?

• No, the function is not differentiable at $x = 1$ because there is a sharp corner, indicating that the derivative does not exist at that point.

Would you like more detailed explanations or clarification on any of the points?

Here are some related questions to explore further:

1. How do we determine whether a function is differentiable at a point?
2. What role do limits play in determining continuity?
3. Can a function be continuous but not differentiable? Why?
4. What graphical features typically indicate non-differentiability?
5. What is the difference between a corner and a cusp in a graph?

Tip: Differentiability implies continuity, but continuity does not imply differentiability. Always check for sharp points or cusps when determining differentiability.