Q1. Analyze the function f(x) shown in the graph:

• Find the following limits:

• State all horizontal asymptotes for . If there are none, state that.

• State all vertical asymptotes for . If there are none, state that.

• Identify all points of discontinuity for AND indicate the type of discontinuity for each one.

• At which points, if any, is continuous but not differentiable?

Background

Topic: Limits, Continuity, and Asymptotes from a Graph

This question tests your ability to interpret a graph to determine limits, points of continuity/discontinuity, and the existence of asymptotes. You will also need to identify where the function is continuous but not differentiable.

Key Terms and Formulas

• Limit: is the value that approaches as gets close to .

• One-sided limit: (from the left), (from the right).

• Horizontal asymptote: A line where or .

• Vertical asymptote: A line where increases or decreases without bound as approaches .

• Discontinuity: A point where the function is not continuous. Types include jump, removable, and infinite discontinuities.

• Continuous but not differentiable: Points where the function is continuous but has a sharp corner or cusp.

Step-by-Step Guidance

1. Examine the graph at each specified -value to determine the behavior of as $x$ approaches those points. Look for open/closed circles, jumps, or vertical behavior.

2. For each limit, check if the function approaches the same value from both sides (for two-sided limits) or from the specified direction (for one-sided limits).

3. Identify horizontal asymptotes by observing the end behavior of the graph as and . Does approach a constant value?

4. Identify vertical asymptotes by finding -values where the graph shoots up or down without bound (often where the function is undefined).

5. Locate points of discontinuity by looking for jumps, holes (open circles), or vertical asymptotes. Classify each as jump, removable, or infinite discontinuity.

6. Check for points where the graph is continuous but has a sharp corner or cusp—these are points where the function is not differentiable.

Try solving on your own before revealing the answer!