The image contains a problem asking to evaluate the limit of functions as $x \to 0$. Here's a breakdown of the tasks:

Problem Statement

• Instruction: "Estimate the limit of $f(x)$ as $x \to 0$ for the following functions (if they exist). Use a calculator if necessary."
• Notation: $|a|$ represents the greatest integer less than or equal to $a$.

Functions to analyze:

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

   • Example provided: $\left\lfloor 2.7 \right\rfloor = 2$.

2. (b) $f(x) = x \left\lceil \frac{1}{x} \right\rceil$, where $\left\lceil a \right\rceil$ represents the smallest integer greater than or equal to $a$.

   • Example provided: $\left\lceil 2.3 \right\rceil = 2$.

3. (c) $f(x) = |x|$, where $|x|$ represents the absolute value of $x$.

Solutions:

(a) $f(x) = x \left\lfloor \frac{1}{x} \right\rfloor$

For this function, as $x \to 0$ from the positive side, $\frac{1}{x} \to \infty$, and $\left\lfloor \frac{1}{x} \right\rfloor$ will be a large integer. So, $f(x) = x \left\lfloor \frac{1}{x} \right\rfloor.$ For small positive $x$, the value $\left\lfloor \frac{1}{x} \right\rfloor$ will be close to $\frac{1}{x}$, making $f(x)$ approximately 1.

When $x \to 0$ from the negative side, $\frac{1}{x} \to -\infty$, and $\left\lfloor \frac{1}{x} \right\rfloor$ will be a large negative integer. So, $f(x)$ will be close to -1.

Thus, the limit does not exist, as the behavior differs from the left and the right.

(b) $f(x) = x \left\lceil \frac{1}{x} \right\rceil$

For small positive $x$, $\left\lceil \frac{1}{x} \right\rceil \approx \frac{1}{x}$, so $f(x) \approx 1$.

For small negative $x$, $\left\lceil \frac{1}{x} \right\rceil \approx \frac{1}{x}$, which gives $f(x) \approx -1$.

Again, the limit does not exist as $x \to 0$, since the values from the left and the right differ.

(c) $f(x) = |x|$

The function $f(x) = |x|$ is continuous and well-behaved. As $x \to 0$, the value of $|x| \to 0$.

Thus, the limit is: $\lim_{x \to 0} |x| = 0.$

Summary:

• For (a), the limit does not exist.
• For (b), the limit does not exist.
• For (c), the limit is 0.

Would you like further clarification on any of these steps or explanations?

Follow-up questions:

1. Why does the floor function behave differently for positive and negative values of $x$?
2. How does the ceiling function affect the limit as $x \to 0$?
3. Can you think of a modification to $f(x)$ in parts (a) or (b) to make the limit exist?
4. What is the significance of the absolute value in part (c)?
5. Can the limit of $f(x) = x \cdot g(x)$ exist if $g(x)$ oscillates rapidly?

Tip:

Always check the behavior of the function from both the left and right sides when computing limits at a point of discontinuity.