The document contains a homework assignment with the following math problems:

Question 1:

Sketch the graph of a function $y = f(x)$ that has the following properties:

• $\lim_{x \to \infty} f(x) = -3$
• $\lim_{x \to -1^+} f(x) = -\infty$
• $f(-1) = 6$
• $\lim_{x \to -\infty} f(x) = 2$
• $\lim_{x \to -1^-} f(x) = 4$
• $\lim_{x \to 3} f(x) = 2$
• $f(3) = -1$

To sketch this graph, consider the behavior at specific points and limits:

• As $x \to \infty$, the function approaches $-3$, suggesting a horizontal asymptote at $y = -3$.
• As $x \to -1^+$, the function approaches $-\infty$, indicating a vertical asymptote at $x = -1$ from the right.
• At $x = -1$, the function value jumps to 6.
• As $x \to -\infty$, the function approaches 2.
• From the left side of $x = -1$, the function approaches 4, indicating a discontinuity at $x = -1$.
• As $x \to 3$, the function approaches 2, but $f(3) = -1$, showing a jump discontinuity at $x = 3$.

Question 2:

Evaluate the following limit: $\lim_{x \to 1} \left[ (x - 1) \sin\left(2 \sqrt{x}\right) \right]$ This is a limit expression involving a product of $(x - 1)$ and a sine function. To solve this:

• Analyze the behavior of each term as $x \to 1$.
• Use limit laws and potentially apply L'Hopital's Rule if the expression results in an indeterminate form such as $0 \cdot 0$.

Would you like me to go into further detail for solving these questions?

---

Here are 5 follow-up questions:

1. What does a vertical asymptote imply about the function's behavior at a specific point?
2. How do jump discontinuities affect the continuity of a function?
3. What is the significance of using L'Hopital's Rule in limits?
4. How does the sine function behave near 0?
5. Can a function have more than one horizontal asymptote?

Tip: When sketching graphs, always mark critical points, asymptotes, and discontinuities to ensure accuracy.