https://file.aiquickdraw.com/mathbot/file/172574344095725htzuyp.pdf

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?

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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.

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].

Learn how to sketch graphs of functions with limits, asymptotes, and discontinuities. This problem explores the behavior of functions as they approach infinity, discontinuities at specific points, and limit calculations using L'Hopital's Rule. Suitable for advanced high school students.

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