#🔗 1.9 Connecting Multiple Representations of Limits

Hey there, future AP Calc master! 👋 This section is all about bringing together everything you've learned about limits. We're going to see how graphs, tables, and equations all tell the same story, just in different ways. Think of it like being a detective, using all the clues to solve the mystery of the limit! 🕵️‍♀️

#🧮 Connecting the Dots: Graphs, Tables, and Equations

Key Concept

Connecting different representations of limits is a huge deal on the AP exam. It shows you really understand what a limit is, not just how to calculate it. 💡

To ace this, you've got to be able to pull information from each representation and see how they match up. It's like having multiple puzzle pieces that all fit together to create a full picture of the function's behavior near a certain point.

For example, you might use a table to get a rough idea of the limit, then check your answer by looking at the graph or finding the exact limit using algebra. Or you could start with an equation, sketch a graph, and then create a table to confirm your findings. It's all about flexibility! 🔄

Sometimes, one representation is more helpful than the others. If an equation is a mess to work with, a graph might give you the insight you need. Let's quickly review each representation:

Numerical (Tables) 🔢
- Plug in values of x that get closer and closer to your target value.
- Look at what the f(x) values are approaching. This gives you an approximation, not the exact limit.
Graphical 📊
- Look at the graph and see where the function is headed as x approaches your target value from both sides.
- Like tables, this is an approximation, not the exact limit.
Algebraic 📝
- Use factoring, rationalizing, or trig identities to simplify the function.
- This method can give you the exact limit.

Let's dive into an example!

#✏️ Example: Multiple Representations of Limits

Suppose we have a function f where $\lim_{x \to 0}f(x) = 1$ . Which of the following could represent f?

Option A)

Piecewise Function

Note: There's a removable discontinuity at (0, 1). We'll cover this in more detail later.

Option B)

x	-0.2	-0.1	-0.001	0.001	0.1	0.2
f(x)	1.04	1.01	1.000001	1.000001	1.01	1.04

Option C)

$f(x)=\begin{cases} x-5, x<0\\ 1, x\geq 0\end{cases}$

Here’s how we solve this:

Graph (Option A): As x approaches 0 from the left, f(x) approaches 1. But as x approaches 0 from the right, f(x) approaches -3. Since the left and right limits are different, option A is incorrect.
Table (Option B): As x gets closer to 0 from both sides, f(x) gets closer to 1. So, this could be a correct representation.
Equation (Option C): For x values approaching 0 from the left, f(x) approaches 0 - 5 = -5. For x values approaching 0 from the right, f(x) is 1. Since the left and right limits are different, option C is incorrect.

So, the correct answer is Option B.

#🚀 Your Turn: Limits Practice Problem

Time to test your skills! 🎯

#❓ Practice Question

Let f be a function where $\lim_{x \to 4}f(x) = 5$ . Which of the following could represent f?

a) $f(x)=\begin{cases} \frac{x^2-3x-4}{x-4}, x \neq 4\\ 6, x = 4\end{cases}$

b)

Piecewise Function

c)

x	3.8	3.9	3.999	4	4.001	4.01	4.02
f(x)	6.2	6.01	6.001	4	4.999	4.9	4.8

#✅ Practice Solution

Let's break it down:

Equation (Option a): We can simplify the equation:

$f(x)=\begin{cases} \frac{(x+1)(x-4)}{x-4}, x \neq 4\\ 6, x = 4\end{cases}$

For x ≠ 4, the function simplifies to f(x) = x + 1. Plugging in x = 4, we get 5. So, this could be the correct answer.
Graph (Option b): As x approaches 4 from the left, f(x) approaches 5. But as x approaches 4 from the right, f(x) approaches 6. So, this is incorrect.
Table (Option c): As x approaches 4 from the left, f(x) approaches 6. As x approaches 4 from the right, f(x) approaches 5. Since the left and right limits are different, this is incorrect.

Therefore, the correct answer is Option a.

Practice Question

Multiple Choice

The function f is defined by $f(x) = \frac{x^2 - 4}{x - 2}$ for $x \ne 2$ . What value should be assigned to $f(2)$ to make f continuous at x = 2?

(A) 0 (B) 1 (C) 2 (D) 4 (E) The function cannot be continuous at x=2
Given the function g(x), what is $\lim_{x \to 3} g(x)$ if $g(x) = \begin{cases} x^2 - 2, & x < 3 \\ 4x - 5, & x \geq 3 \end{cases}$ ?

(A) 7 (B) 11 (C) The limit does not exist (D) 4 (E) 1

Free Response

Consider the function $h(x)$ defined as follows:

$h(x) = \begin{cases} ax^2 + b, & x \leq 1 \\ 3x - 2, & 1 < x \leq 3 \\ c \sqrt{x} + 1, & x > 3 \end{cases}$

(a) Find the values of a and b such that $h(x)$ is continuous at x = 1. (b) Find the value of c such that $h(x)$ is continuous at x = 3. (c) Is $h(x)$ differentiable at x = 1 and x = 3? Justify your answer.

Solution:

(a) For continuity at x=1: - $a(1)^2 + b = 3(1) - 2$ - $a + b = 1$

Since we need more information to solve for both *a* and *b*, we can only state that the relationship between *a* and *b* is *a + b = 1*.

(b) For continuity at x=3: - $3(3) - 2 = c \sqrt{3} + 1$ - $9 - 2 = c \sqrt{3} + 1$ - $7 = c \sqrt{3} + 1$ - $6 = c \sqrt{3}$ - $c = \frac{6}{\sqrt{3}} = 2\sqrt{3}$

(c) - Differentiability at x=1 requires the left and right derivatives to be equal. The left derivative is $2ax$ which at $x=1$ is $2a$ . The right derivative is 3. So, for differentiability, $2a = 3$ or $a = 3/2$ . Since we know that $a + b = 1$ , we can find $b = -1/2$ . So, h(x) is differentiable at x = 1 if and only if a = 3/2 and b = -1/2. - Differentiability at x=3 requires the left and right derivatives to be equal. The left derivative is 3. The right derivative is $\frac{c}{2\sqrt{x}}$ which at $x=3$ is $\frac{c}{2\sqrt{3}} = \frac{2\sqrt{3}}{2\sqrt{3}} = 1$ . Since the left and right derivatives are not equal, h(x) is not differentiable at x = 3.

#🎉 Wrapping Up

Great job! You've now seen how to connect different representations of limits. This skill is crucial for the AP exam, where you'll often need to combine different approaches to solve problems. You're doing awesome! Keep up the great work! 💪

This topic is a high-value topic because it combines all the limit concepts you've learned so far. Expect to see this kind of question frequently on the AP exam. 💯