In the following exercises, sketch the graph of a function with the given properties.
As x→−∞, f(x)→2,limx→3−f(x)=−∞,
limx→3+f(x)=∞, As x→∞, f(x)→2,f(0)=−1/3

Let's analyze the given properties step by step in order to sketch the graph of the function \( f(x) \):

### 1. **As \( x \to -\infty \), \( f(x) \to 2 \):**
   - This means that as \( x \) moves towards negative infinity, the function approaches the value 2. So, the left tail of the graph will flatten out toward \( y = 2 \).

### 2. **\( \lim_{x \to 3^-} f(x) = -\infty \):**
   - As \( x \) approaches 3 from the left, the function goes to negative infinity. This indicates a vertical asymptote at \( x = 3 \), where the function plunges downwards as it gets closer to 3 from the left side.

### 3. **\( \lim_{x \to 3^+} f(x) = \infty \):**
   - As \( x \) approaches 3 from the right, the function shoots up towards positive infinity. This means that after crossing \( x = 3 \), the function ascends rapidly, reinforcing the vertical asymptote at \( x = 3 \).

### 4. **As \( x \to \infty \), \( f(x) \to 2 \):**
   - As \( x \) moves towards positive infinity, the function approaches the value 2 again. So, the right tail of the graph will also flatten out towards \( y = 2 \).

### 5. **\( f(0) = -\frac{1}{3} \):**
   - At \( x = 0 \), the function value is \( -\frac{1}{3} \). This gives a specific point on the graph: \( (0, -\frac{1}{3}) \).

### Summary of the Graph:
- **Horizontal asymptotes**: At \( y = 2 \) as \( x \to \pm\infty \).
- **Vertical asymptote**: At \( x = 3 \), with \( f(x) \to -\infty \) as \( x \to 3^- \) and \( f(x) \to \infty \) as \( x \to 3^+ \).
- **Point**: \( (0, -\frac{1}{3}) \) is on the graph.

The graph will likely approach the horizontal asymptote \( y = 2 \) at both extreme ends (negative and positive infinity), and there will be a sharp drop and rise near \( x = 3 \) due to the vertical asymptote.

### Would you like me to draw this graph for you?

#### Five related questions:
1. What is the significance of horizontal asymptotes in graphing functions?
2. How can vertical asymptotes affect the behavior of a graph?
3. What are the general steps to find limits of a function as \( x \to \infty \)?
4. How do you determine if a function has a jump discontinuity or infinite discontinuity?
5. Can a function have both vertical and horizontal asymptotes at the same point?

### Tip:
When sketching a function with asymptotes, it's useful to mark the asymptotes first and then sketch the behavior of the graph around them.

In the following exercises, sketch the graph of a function with the given properties. As x→−∞, f(x)→2, limx→3−f(x)=−∞, limx→3+f(x)=∞, As x→∞, f(x)→2, f(0)=−1/3.

Learn how to sketch the graph of a function given specific limit and asymptote properties. This example covers limits at infinity, vertical and horizontal asymptotes, and function behavior at key points. Suitable for students in Grades 11-12 studying precalculus or calculus.

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