To devise a function $f(x)$ that meets the conditions given in the problem:

Function Construction:

You want to create a function $f(x)$ such that:

• $\lim\limits_{x \to 2^-} f(x) = 4$
• $\lim\limits_{x \to 2^+} f(x) = 9$

One simple way to achieve this is to define a piecewise function:

\begin{cases} 4 & \text{if } x < 2 \\ 9 & \text{if } x > 2 \end{cases}$$ This function will give you the desired limits: - As $$x$$ approaches 2 from the left ($$x \to 2^-$$), $$f(x)$$ is always 4, so $$\lim\limits_{x \to 2^-} f(x) = 4$$. - As $$x$$ approaches 2 from the right ($$x \to 2^+$$), $$f(x)$$ is always 9, so $$\lim\limits_{x \to 2^+} f(x) = 9$$. ### Regarding the Existence of $$\lim\limits_{x \to 2} f(x)$$: For the limit $$\lim\limits_{x \to 2} f(x)$$ to exist, the left-hand limit ($$\lim\limits_{x \to 2^-} f(x)$$) and the right-hand limit ($$\lim\limits_{x \to 2^+} f(x)$$) must be equal. In this case: - $$\lim\limits_{x \to 2^-} f(x) = 4$$ - $$\lim\limits_{x \to 2^+} f(x) = 9$$ Since 4 is not equal to 9, the two-sided limit $$\lim\limits_{x \to 2} f(x)$$ **does not exist**. ### Final Answer: - **True or False**: Is it possible for $$\lim\limits_{x \to 2} f(x)$$ to exist? - **False**. The limit does not exist because the one-sided limits are not equal. Would you like to explore more about this or need further clarification? ### Follow-up Questions: 1. What conditions must be met for a two-sided limit to exist? 2. How would the function change if both one-sided limits were equal? 3. Can you create another piecewise function that meets the same limit conditions but is defined differently? 4. What happens to the graph of $$f(x)$$ at $$x = 2$$ based on this function? 5. How does the concept of limits help in understanding the behavior of functions near discontinuities? ### Tip: Always check both the left-hand and right-hand limits before concluding the existence of a two-sided limit.