The question in the image asks to estimate the limit of the given functions as $x$ approaches 0. The functions are defined as follows:

1. For part (a): $f(x) = \frac{1}{x} \left\lfloor \frac{1}{x} \right\rfloor$ where $\lfloor \alpha \rfloor$ represents the greatest integer less than or equal to $\alpha$.

2. For part (b): $f(x) = x \left\lfloor \frac{1}{x} \right\rfloor$ where $\lfloor \alpha \rfloor$ has the same meaning as before.

3. For part (c): $f(x) = \lfloor x \rfloor$ where $\lfloor x \rfloor$ is the floor function.

Let's analyze each part:

Part (a)

The expression $\frac{1}{x} \left\lfloor \frac{1}{x} \right\rfloor$ can be tricky because as $x$ approaches 0, $\frac{1}{x}$ becomes very large. The floor function $\lfloor \frac{1}{x} \rfloor$ also approaches a large integer, but its exact value depends on whether $x$ is positive or negative. Thus, as $x$ gets closer to 0 from either direction, this expression doesn't approach a single value, indicating that the limit does not exist.

Part (b)

The function $f(x) = x \left\lfloor \frac{1}{x} \right\rfloor$ is interesting because as $x$ approaches 0, the product involves $x$ multiplying the floor of a large number. This form often tends towards 0 since $x$ becomes smaller faster than $\left\lfloor \frac{1}{x} \right\rfloor$ becomes larger. Hence, the limit as $x$ approaches 0 is most likely 0.

Part (c)

For the function $f(x) = \lfloor x \rfloor$, as $x$ approaches 0 from the right (positive side), the floor value is 0, and as $x$ approaches from the left (negative side), the floor value is -1. Therefore, the limit does not exist since the left-hand limit and right-hand limit are not equal.

Summary of Limits:

• (a): The limit does not exist.
• (b): The limit is 0.
• (c): The limit does not exist.

Would you like more details on any of these parts or have any further questions?

Here are some related questions you might consider exploring:

1. What is the formal definition of a limit and how does it apply to these cases?
2. How does the behavior of the floor function influence the outcome of limits?
3. What happens to the expressions if we consider one-sided limits instead of two-sided limits?
4. Can you think of a similar function to part (b) whose limit does not exist?
5. How would the answers change if we replaced the floor function with the ceiling function?

Tip: When dealing with limits involving floor functions, always check the behavior as you approach from both sides (positive and negative) to see if the limits agree.