Lab: PLACEHOLDER Graphical Exploration of Limits

Objective

Develop an intuitive understanding of limits by exploring function behavior near specific points using graphical analysis.

Materials Needed

• Graphing calculator or computer with graphing software
• Graph paper
• Ruler
• Magnifying glass (optional, for detailed graph analysis)

Part 1: The Mystery of the "Hole"

Time: 15 minutes

Consider the function: $$f(x) = \frac{x^2 - 4}{x - 2}$$

Investigation A: What happens at x = 2?

1. Try to evaluate $f(2)$ directly. What happens?

2. Create a table of values approaching $x = 2$:

3. Graph the function on your calculator. What do you observe at $x = 2$?

4. Zoom in around $x = 2$. What does the graph suggest about the "missing" point?

Discovery Questions:

• What value does $f(x)$ seem to approach as $x$ gets close to 2?
• Does it matter whether $x$ approaches 2 from the left or right?
• How would you "fill in the hole" in the graph?

Part 2: One-Sided Limits

Time: 15 minutes

Investigation B: The Step Function

Consider the piecewise function:

$$g(x) = \begin{cases} x + 1 & \text{if } x < 1 \\ 3 & \text{if } x = 1 \\ 2x & \text{if } x > 1 \end{cases}$$

1. Graph this function carefully. Pay special attention to what happens at $x = 1$.

2. Trace along the curve approaching $x = 1$ from the left. What y-value do you approach?

3. Trace along the curve approaching $x = 1$ from the right. What y-value do you approach?

4. What is the actual value of $g(1)$?

Analysis Table:

Discovery Questions:

• Do the left and right approaches give the same result?
• Does the limit exist at $x = 1$? Why or why not?

Part 3: Infinite Behavior

Time: 10 minutes

Investigation C: Vertical Asymptotes

Explore the function: $$h(x) = \frac{1}{x - 3}$$

1. What happens when you try to evaluate $h(3)$?

2. Create approach tables:

From the left of x = 3:

From the right of x = 3:

3. Graph the function and observe the behavior near $x = 3$.

Discovery Questions:

• What happens to the y-values as x approaches 3 from each side?
• How would you describe this behavior mathematically?
• What does "limit equals infinity" mean?

Part 4: Oscillating Functions

Time: 10 minutes

Investigation D: When Limits Don't Exist

Examine the function: $$k(x) = \sin\left(\frac{1}{x}\right)$$ near $x = 0$.

1. Graph this function using a viewing window like $[-0.5, 0.5]$ by $[-2, 2]$.

2. Zoom in progressively closer to $x = 0$. What do you observe?

3. Try to trace the function as it approaches $x = 0$. What happens?

Discovery Questions:

• Does the function settle on a single value as $x$ approaches 0?
• How would you describe the function's behavior near $x = 0$?
• Why might this limit not exist?

Part 5: Synthesis Activity

Time: 10 minutes

Your Turn: Function Detective

For each graph below, determine if the limit exists at the indicated point. If it does, estimate the limit value.

Function A: (Draw or describe a function with a removable discontinuity)

• $\lim_{x \to 2} A(x) = $ _______

Function B: (Draw or describe a function with a jump discontinuity)

• $\lim_{x \to -1} B(x) = $ _______

Function C: (Draw or describe a function with a vertical asymptote)

• $\lim_{x \to 0} C(x) = $ _______

Part 6: Real-World Connection

Time: 5 minutes

Consider the cost function for a shipping company:

$$C(w) = \begin{cases} 5.00 & \text{if } 0 < w \leq 1 \\ 8.00 & \text{if } 1 < w \leq 2 \\ 11.00 & \text{if } 2 < w \leq 3 \end{cases}$$

where $w$ is weight in pounds.

1. What is $\lim_{w \to 1^-} C(w)$?
2. What is $\lim_{w \to 1^+} C(w)$?
3. Does $\lim_{w \to 1} C(w)$ exist?
4. What does this mean in practical terms for shipping costs?

Reflection and Conclusions

Key Discoveries

Write a brief explanation of each concept:

1. Removable Discontinuity: _________________________________

2. Jump Discontinuity: _____________________________________

3. Infinite Limits: _______________________________________

4. Oscillating Behavior: __________________________________

The Big Picture

In your own words, explain what a limit represents and why it's useful in mathematics.

Questions for Further Exploration

1. How might limits help us understand rates of change?
2. What happens to limits when we have functions like $\frac{\sin x}{x}$ as $x \to 0$?
3. How do limits relate to the idea of continuity?

Lab Report

Create a visual summary showing:

• One example each of the four types of limit behavior you explored
• A written explanation of when limits exist vs. when they don't
• One real-world situation where understanding limits would be important

Extension Challenge

Design your own piecewise function that has:

• A removable discontinuity at $x = 1$
• A jump discontinuity at $x = 2$
• A vertical asymptote at $x = 3$

Graph your function and verify that it has these properties.