To solve the given problem, let's break down each part of the question.

We are given that $f(x) = \lfloor x \rfloor + \lfloor -x \rfloor$, where $\lfloor x \rfloor$ represents the greatest integer less than or equal to $x$.

The goal is to show that the limit $\lim_{x \to 2} f(x)$ exists, but it is not equal to $f(2)$.

Step 1: Understanding $f(x)$

The function is defined as: $f(x) = \lfloor x \rfloor + \lfloor -x \rfloor$

For any integer $a$, $f(a) = \lfloor a \rfloor + \lfloor -a \rfloor$. Now, let's explore the function behavior near $x = 2$.

Step 2: Evaluating the limit $\lim_{x \to 2^-} f(x)$

For $x \to 2^-$ (values of $x$ slightly less than 2), we have:

• $\lfloor x \rfloor = 1$ (since $x < 2$ but close to 2),
• $\lfloor -x \rfloor = -2$ (since $-x$ is slightly greater than -2).

Thus, $f(x) = 1 + (-2) = -1$ when $x \to 2^-$.

$\lim_{x \to 2^-} f(x) = -1$

Step 3: Evaluating the limit $\lim_{x \to 2^+} f(x)$

For $x \to 2^+$ (values of $x$ slightly greater than 2), we have:

• $\lfloor x \rfloor = 2$ (since $x$ is now just above 2),
• $\lfloor -x \rfloor = -3$ (since $-x$ is slightly less than -2).

Thus, $f(x) = 2 + (-3) = -1$ when $x \to 2^+$.

$\lim_{x \to 2^+} f(x) = -1$

Step 4: Conclusion about the limit

Since both the left-hand limit and the right-hand limit agree, we can conclude: $\lim_{x \to 2} f(x) = -1$

Step 5: Calculating $f(2)$

At $x = 2$:

• $\lfloor 2 \rfloor = 2$,
• $\lfloor -2 \rfloor = -2$.

Thus, $f(2) = 2 + (-2) = 0$.

Step 6: Final Conclusion

We have shown that: $\lim_{x \to 2} f(x) = -1 \quad \text{and} \quad f(2) = 0$

Therefore, $\lim_{x \to 2} f(x) \neq f(2)$.

Would you like a further detailed explanation or clarification on any of the steps? Here are five follow-up questions related to this:

1. What happens to the function $f(x)$ near other integer points?
2. How does the behavior of floor functions impact discontinuities in functions like this?
3. Why are there holes at integer points in the graph of $f(x)$?
4. Can we generalize this limit behavior to other values of $x$ where $x$ is an integer?
5. How does the floor function handle negative values differently from positive values?

Tip: When evaluating limits involving piecewise or discontinuous functions, always check both the left-hand and right-hand limits to see if they match.