#AP Calculus AB/BC: Algebraic Manipulation for Limits 🚀

Hey there, future AP Calculus master! Let's get you prepped to ace those limit problems using some awesome algebraic tricks. This guide is designed to be your go-to resource, especially the night before the exam. We'll break down everything step-by-step, making sure you're confident and ready to rock!

This section covers essential techniques that appear frequently on the AP exam. Mastering these methods will significantly boost your score.

# 1. Factoring: Your First Line of Attack 🛠️

Factoring is like breaking down a complex problem into smaller, easier-to-handle pieces. It's all about finding common factors to simplify expressions.

#How It Works:

• Look for common factors in all terms of the expression.
• Factor out the greatest common factor (GCF).
• Simplify the expression as much as possible.

#🚆 Example Set 1: Factor Away!

#Example Set 1: Question 1

Find the limit:

$$\lim_{x \rightarrow 3} (x^{3} - x^{2} - x)$$

Solution:

1. Factor out x: $x(x^2 - x - 1)$
2. Plug in x = 3: $3(3^2 - 3 - 1) = 3(9 - 3 - 1) = 3(5) = 15$

#Example Set 1: Question 2

Find the limit:

$$\lim_{x \rightarrow -1} (\frac{x^{3} - 2x^{2}}{3})$$

Solution:

1. Factor out $x^2$: $x^2(\frac{x-2}{3})$
2. Plug in x = -1: $(-1)^2(\frac{-1-2}{3}) = 1(\frac{-3}{3}) = -1$

#Example Set 1: Question 3

Find the limit:

$$\lim_{x \rightarrow 1} (\frac{x^{2} - 5x - 6}{x^{2} - 7x + 6})$$

Solution:

1. Factor the numerator: $(x + 1)(x - 6)$
2. Factor the denominator: $(x - 1)(x - 6)$
3. Simplify: $\frac{(x+1)}{(x-1)}$
4. Plug in x = 1: $\frac{2}{0}$, which means the limit does not exist.