Let's analyze the graph of $y = f(x)$ and answer each of the provided questions:

(a) Does $\lim_{x \to 0} f(x)$ exist?

From the graph, observe the behavior of $f(x)$ as $x$ approaches 0 from both sides (left and right):

• As $x$ approaches 0 from the left ($x \to 0^-$), the function decreases without bound, tending towards negative infinity.
• As $x$ approaches 0 from the right ($x \to 0^+$), the function increases without bound, tending towards positive infinity.

Since the left-hand and right-hand limits are not equal and both tend towards infinity in opposite directions, the limit does not exist.

Answer: No

(b) Is $f(x)$ continuous at $x = 0$?

For a function to be continuous at a point, three conditions must be satisfied:

1. The function must be defined at that point.
2. The limit as $x$ approaches the point must exist.
3. The value of the function at the point must equal the limit as $x$ approaches the point.

From the graph, we see that:

• The limit as $x \to 0$ does not exist.
• Additionally, it seems that $f(0)$ is undefined, as there is no specific point shown for $x = 0$.

Therefore, $f(x)$ is not continuous at $x = 0$.

Answer: No

(c) Is $f(x)$ differentiable at $x = 0$?

A function is differentiable at a point if it is continuous and has a well-defined slope (tangent) at that point. Since we already determined that $f(x)$ is not continuous at $x = 0$, it cannot be differentiable there.

Answer: No

Let me know if you'd like more details or have any questions! Here are some related questions to explore further:

1. What are the necessary conditions for differentiability at a point?
2. Can a function be continuous but not differentiable at a point? If so, give an example.
3. What is the relationship between left-hand and right-hand limits for continuity?
4. How does the existence of vertical asymptotes affect the limit at a point?
5. What are common examples of functions with discontinuities at a point?

Tip: Always check the behavior of the function from both sides when evaluating limits!