This problem asks you to sketch the graph of a function $y = f(x)$ that satisfies several given conditions. Let's break down these conditions and discuss how the graph might behave.

Conditions and Interpretation:

1. For all $x < 0$, $\lim_{t \to x} f(t) = f(x)$:

   • This means the function is continuous for all $x < 0$, meaning no breaks or jumps in this region.

2. $\lim_{x \to -\infty} f(x) = -\infty$:

   • As $x$ becomes more negative, the function decreases without bound.

3. $f(x) \to 4$ as $x \to -\infty$:

   • This condition seems to contradict the previous one; it might be a typo. If we assume this means "as $x \to \infty$, $f(x) \to 4$," then the function should approach $y = 4$ as $x \to positive infinity$.

4. $f(2) = 3$:

   • The value of the function at $x = 2$ is exactly 3.

5. $f$ is not continuous at $x = 2$:

   • This indicates there is a discontinuity at $x = 2$, possibly a jump discontinuity.

6. $\lim_{x \to 2^-} f(x) = 3$:

   • As $x$ approaches 2 from the left, the function approaches the value 3.

Steps to Sketch the Graph:

• For $x < 0$:

  • The function is continuous.
  • As $x \to -\infty$, the function should decrease without bound.
  • As $x \to 0$, it remains continuous.

• For $x \to \infty$:

  • The function should approach $y = 4$, as mentioned.

• At $x = 2$:

  • The function takes the value $3$ at this point, but since it's not continuous, there will be a jump or break. Specifically, the left-hand limit at 2 is 3, but the right-hand limit may be different or undefined.

Would you like a sketch based on this interpretation or further clarification on any point?

Relative Questions:

1. How does the graph behave for values of $x > 2$?
2. What would happen if the function were continuous at $x = 2$?
3. How do left-hand and right-hand limits affect the continuity of a function?
4. What type of discontinuity could occur at $x = 2$ (e.g., jump, infinite, or removable)?
5. How can we determine the behavior of a function as $x \to \pm \infty$?

Tip: When graphing, ensure to label key points such as discontinuities and asymptotic behavior for clarity.