To analyze the problem provided in the image, let us break it down step by step:

1. Understanding the Problem:

   • You are tasked to identify which functions, among the provided graphs, have a defined limit as $x \to 1$.
   • The condition for a limit at $x = 1$ is $\lim_{x \to 1} f(x) = a$, where $a \in \mathbb{R}$, meaning the left-hand limit ($x \to 1^-$) and right-hand limit ($x \to 1^+$) must be equal and finite.

2. Analysis of Each Graph:

   • First Graph: The function exhibits oscillations near $x = 1$, indicating that the left-hand limit and right-hand limit may not coincide. Therefore, the limit does not exist.
   • Second Graph: The function approaches negative infinity as $x \to 1^-$ and positive infinity as $x \to 1^+$. Since the limits are not equal and not finite, the limit does not exist.
   • Third Graph: Both left-hand and right-hand limits approach the same finite value at $x = 1$. Thus, the limit exists.
   • Fourth Graph: Similar to the third graph, both left-hand and right-hand limits converge to the same finite value at $x = 1$. Hence, the limit exists.

3. Conclusion:

   • The third and fourth graphs satisfy the condition $\lim_{x \to 1} f(x) = a$.

Would you like me to elaborate on any part of the analysis or provide additional examples?

Related Questions:

1. What are the conditions for the existence of a limit at a point for a function?
2. How does continuity relate to the existence of limits?
3. Can a function have a limit at a point where it is not defined? Why?
4. How do you distinguish between left-hand and right-hand limits graphically?
5. What happens if a function oscillates infinitely as $x \to c$?

Tip:

When checking for limits graphically, focus on whether the left-hand and right-hand values align at the target point. Discontinuities, jumps, or infinite oscillations typically indicate the absence of a limit.