Lesson 2.4

The Leading Coefficient Test

We've used words like "Up" and "Down." Now, we'll use the precise language of Calculus: Limits.

Introduction

Is there a way to describe "starts low, ends high" using math symbols? Yes. We talk about where goes as goes to infinity.

Past Knowledge

You know (x goes right) and (x goes left).

Today's Goal

Combine Degree (Even/Odd) and Leading Coefficient (+/-) into 4 distinct cases using formal notation.

Future Success

Limit notation is the FIRST topic in Calculus. Mastering it now puts you way ahead.

Key Concepts

1. The Notation

We rewrite visual descriptions into math sentences.

"Right Side goes UP"

"Left Side goes DOWN"

2. The Grid

Degree \ Sign	Positive (+)	Negative (-)
Even	UP / UP	DOWN / DOWN
Odd	DOWN / UP	UP / DOWN

Worked Examples

Example 1: Translating to Limits

Basic

Use limits to describe .

Identify Shape

Degree 4 (Even) + Negative Coeff (-2) = Down / Down.

Write Limits

Left Side:

Right Side:

Example 2: Reverse Engineer

Concept

If and , what do we know?

1. Right goes UP ()

2. Left goes DOWN ()

This is "Down / Up".

So the function must have an Odd Degree and a Positive Leading Coefficient.

Example 3: Complex Analysis

Advanced

Determine the end behavior limits for .

Distribute or Analyze Logic

The leading term comes from multiplying the biggest parts:
.

State the Limits

Even Degree + Positive LC = Up / Up.

Common Pitfalls

x vs f(x)

describes Left/Right. describes Up/Down. Don't swap them!

The Arrows

The arrow () means "approaches". It never equals infinity; it just gets closer and closer.

Real-Life Applications

In algorithmic trading, knowing the "limit at infinity" tells you the long-term trend of a stock model. If the limit is negative infinity, the company effectively goes bankrupt in the long run, no matter how good the short-term bumps are!