We need to sketch a function $f(x)$ that meets the following requirements, including limits and a discontinuity:

1. $\lim_{x \to -2^-} f(x) = 4$ (the left-hand limit at $x = -2$ is 4).
2. $\lim_{x \to -2^+} f(x) = -2$ (the right-hand limit at $x = -2$ is -2).
3. $f(-2) = -3$ (the function value at $x = -2$ is -3, not the same as either of the limits, which indicates a jump discontinuity at $x = -2$).
4. $\lim_{x \to 4} f(x) = -2$ (the function approaches -2 as $x \to 4$).
5. $f(4) = 3$ (the function value at $x = 4$ is 3, indicating another discontinuity, but the limit and the function value are different).

Steps to construct the sketch:

1. At $x = -2$:

   • As $x \to -2$ from the left, $f(x)$ approaches 4. This means as we approach $x = -2$ from the left side, the function should be close to 4.
   • As $x \to -2$ from the right, $f(x)$ approaches -2. The function should approach -2 from the right side of $x = -2$.
   • At $x = -2$, the function value is $f(-2) = -3$, which is neither 4 nor -2, creating a jump discontinuity at $x = -2$. So, at $x = -2$, we place a point at $f(-2) = -3$, and use open circles at 4 (left side) and -2 (right side) to indicate the limits.

2. At $x = 4$:

   • As $x \to 4$, $f(x)$ approaches -2, meaning that as we get close to 4, the function should get close to -2. But the actual function value at $x = 4$ is $f(4) = 3$, indicating another discontinuity at $x = 4$. So, place a point at $f(4) = 3$, and an open circle at $f(x) = -2$ to represent the limit.

3. Continuous behavior elsewhere:

   • Between these two discontinuities at $x = -2$ and $x = 4$, the function must be continuous (no jumps or breaks in the curve), but the specific shape of the curve can vary. It can smoothly connect the required limits and values. A possible simple connection could be a linear or smooth curve transitioning between these points.

Description of the sketch:

• From $x \to -2^-$, the function approaches 4.
• From $x \to -2^+$, the function approaches -2.
• At $x = -2$, the function is $f(-2) = -3$, indicating a jump.
• From $x \to 4$, the function approaches -2.
• At $x = 4$, the function is $f(4) = 3$, again indicating a discontinuity.

Would you like to see a sketch of this function?

---

Related Questions:

1. How does the behavior of limits create discontinuities in a function?
2. What is the difference between a removable discontinuity and a jump discontinuity?
3. Can a function be continuous at a point where its limit exists but differs from the function value?
4. How do left-hand and right-hand limits affect the continuity of a function?
5. What are some other examples of functions with jump discontinuities?

Tip:

When graphing functions with discontinuities, always pay attention to open and closed circles, which indicate where the function has limits versus actual function values.