and
lim x→2+ f(x) = , so
lim x→2 f(x) = . However, f(2) = [[2]] + [[−2]] = 2 +

Let's work through this step by step to show the required properties about the function $f(x) = \lfloor x \rfloor + \lfloor -x \rfloor$, where $\lfloor \cdot \rfloor$ represents the greatest integer function or the floor function.

Step 1: Analyze the behavior of $f(x)$

We know that: $f(x) = \lfloor x \rfloor + \lfloor -x \rfloor$ The function $\lfloor x \rfloor$ returns the greatest integer less than or equal to $x$, and similarly, $\lfloor -x \rfloor$ returns the greatest integer less than or equal to $-x$.

For non-integer values of $x$, this function exhibits a predictable pattern because:

• $\lfloor x \rfloor$ jumps to the next integer as $x$ crosses an integer,
• $\lfloor -x \rfloor$ jumps similarly, but the pattern is inverted.

Step 2: Behavior around $x = 2$

To evaluate the limit, we first check the left-hand and right-hand limits at $x = 2$.

Left-hand limit ($x \to 2^-$):

Consider values of $x$ slightly less than 2, e.g., $x = 1.9$: $f(1.9) = \lfloor 1.9 \rfloor + \lfloor -1.9 \rfloor = \lfloor 1.9 \rfloor + \lfloor -1.9 \rfloor = 1 + (-2) = -1$ This will hold for any $x \in (1, 2)$. Hence: $\lim_{x \to 2^-} f(x) = -1$

Right-hand limit ($x \to 2^+$):

Consider values of $x$ slightly greater than 2, e.g., $x = 2.1$: $f(2.1) = \lfloor 2.1 \rfloor + \lfloor -2.1 \rfloor = \lfloor 2.1 \rfloor + \lfloor -2.1 \rfloor = 2 + (-3) = -1$ This will hold for any $x \in (2, 3)$. Hence: $\lim_{x \to 2^+} f(x) = -1$

Step 3: Conclude the limit exists

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

Step 4: Check the value at $x = 2$

At $x = 2$, we directly compute $f(2)$: $f(2) = \lfloor 2 \rfloor + \lfloor -2 \rfloor = 2 + (-2) = 0$

Step 5: Compare the limit with the value of $f(2)$

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

Conclusion

The limit $\lim_{x \to 2} f(x)$ exists, but it is not equal to $f(2)$, confirming that the limit exists but does not match the function's value at $x = 2$.

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Would you like further details or have any questions?

Here are 5 related questions:

1. What are the properties of the floor function $\lfloor x \rfloor$?
2. Can you generalize this result for other integers $a$ where $x \to a$?
3. How does the behavior of $f(x)$ change as $x$ approaches non-integer values?
4. What is the significance of a function having a limit that differs from its value at a point?
5. What other types of discontinuities exist besides jump discontinuities?

Tip: The floor function creates discontinuities at integer points, leading to potential mismatches between limits and function values at those points.