## Determining Limits Using Algebraic Properties of Limits - AP Calculus Study Guide

### Introduction

Hello math enthusiasts! 🌟 Ready to dive into the world of limits using algebraic properties? It's like cooking, but instead of onions and garlic, we're using nifty algebraic rules. Hold onto your graphing calculators because this is going to be a wild (and educational) ride!

### Discovering the Magic of Limits with Algebra ✨

In calculus, knowing how to find limits isn't just about peeking at a graph and making an educated guess. There's a whole toolkit of algebraic properties that let you find limits quickly and accurately, and today, we're going to unleash those tools! Think of them like math superpowers—maybe not as cool as flying, but definitely useful for passing your AP Calculus exam.

### Key Superpowers: Algebraic Properties of Limits

Let's introduce our limit-finding superheroes: the Sum Rule, Difference Rule, Constant Multiple Rule, Product Rule, Quotient Rule, Power Rule, and Root Rule. Sure, these guys don’t wear capes, but they make calculus a lot easier.

#### Sum Rule 🧮

The Sum Rule states: [ \lim_{{x \to c}} (f(x) + g(x)) = L + M ] Basically, it's telling you that if you know the limits of ( f(x) ) and ( g(x) ) separately, you can just add them together.

**Example:**
Find [ \lim_{{x \to 3}} (x^2 + x^3) ]

The limit of ( x^2 ) as ( x \to 3 ) is 9, and the limit of ( x^3 ) as ( x \to 3 ) is 27. Adding these together using the sum rule, we get 36.

#### Difference Rule ➖

The opposite of the Sum Rule, the Difference Rule tells us: [ \lim_{{x \to c}} (f(x) - g(x)) = L - M ]

**Example:**
Find [ \lim_{{x \to 3}} (x^2 - x^3) ]

Using the limits from the previous example, we subtract them: ( 9 - 27 = -18 ).

#### Constant Multiple Rule 📏

Got a constant hanging out with your function? No problem: [ \lim_{{x \to c}} (k \cdot f(x)) = k \cdot L ]

**Example:**
Find [ \lim_{{x \to 5}} (12x^3) ]

Separate the constant (12) from ( x^3 ), and evaluate the limit of ( x^3 ) as ( x \to 5 ) which is ( 125 ). So, ( 12 \cdot 125 = 1500 ).

#### Product Rule 🍔

Feeling hungry? Try the Product Rule: [ \lim_{{x \to c}} (f(x) \cdot g(x)) = L \cdot M ]

**Example:**
Find [ \lim_{{x \to 5}} (12x^3 \cdot 27x^2) ]

First, find the limits separately. For ( 12x^3 ), it's ( 1500 ). For ( 27x^2 ), plug in 5 to get ( 27 \cdot 25 = 675 ). Then, multiply: ( 1500 \cdot 675 = 1,012,500 ).

#### Quotient Rule ➗

Think of this as the anti-Product Rule: [ \lim_{{x \to c}} \left( \frac{f(x)}{g(x)} \right) = \frac{L}{M}, ; \text{provided that} ; M \neq 0 ]

**Example:**
Find [ \lim_{{x \to 5}} \left( \frac{12x^3}{27x^2} \right) ]

Using the limits we found for ( 12x^3 ) and ( 27x^2 ), divide them: ( \frac{1500}{675} = 2.222 ).

#### Power Rule 💪

The Power Rule tells us about exponents: [ \lim_{{x \to c}} [f(x)]^n = L^n ]

**Example:**
Find [ \lim_{{x \to 5}} (x+4)^3 ]

First, the limit of ( x+4 ) as ( x \to 5 ) is 9. Then cube it: ( 9^3 = 729 ).

#### Root Rule 🌳

Roots are no match for the Root Rule: [ \lim_{{x \to c}} \sqrt[n]{f(x)} = \sqrt[n]{L} ]

**Example:**
Find [ \lim_{{x \to 5}} \sqrt[3]{(x+4)} ]

First, the limit of ( x+4 ) as ( x \to 5 ) is 9. Then take the cube root: (\sqrt[3]{9} \approx 2.08 ).

### Practice Makes Perfect

Now that you know these rules, let's test your skills! Try finding these limits with your new algebraic superpowers:

**Example 1:** [ \lim_{{x \to 2}} (8 - 3x + 12x^2) ]

Plugging in ( 2 ) for ( x ), the equation becomes ( 8 - 3(2) + 12(2^2) = 50 ).

**Example 2:** [ \lim_{{x \to 6}} \frac{x - 3}{x - 3} ]

Plugging in ( 6 ), we get ( \frac{6-3}{6-3} = \frac{3}{3} = 1 ).

**Example 3:** [ \lim_{{x \to 3}} 2^x ]

Using the power rule, we get ( 2^3 = 8 ).

**Example 4:** [ \lim_{{x \to 2}} \sqrt[x]{16} ]

Plugging in ( 2 ), the answer becomes ( 4 ).

### Limits Without a Variable

What if there’s no variable and it’s just constants? Easier than pie! 🍰

**Example 1:** [ \lim_{{x \to 3}} 5\pi ]

Since there’s no variable, it remains ( 5\pi ).

**Example 2:** [ \lim_{{x \to 5}} 2e ]

No variable, so it's just ( 2e ).

### Conclusion

Look at you now, confidently wielding your limit superpowers! Each rule is like a different gadget in Batman’s utility belt—use them wisely, and no limit problem will stand a chance. As you progress in your calculus journey, remember to approach each problem methodically, recognizing the type of expression you're dealing with.

Happy calculating! 🧠🎉