Lesson 21.1

Introduction to Limits

Observing the value a function approaches from the left and right sides of a point.

Introduction

Observing the value a function approaches from the left and right sides of a point.

Past Knowledge

You understand function behavior, asymptotes, and graphing.

Today's Goal

We learn limit notation and evaluate limits from graphs and tables.

Future Success

Limits are the foundation of derivatives, integrals, and all of calculus.

Key Concepts

Limit Notation

As x approaches c, f(x) approaches L.

Left-Hand Limit

Approaching c from values less than c (from the left).

Right-Hand Limit

Approaching c from values greater than c (from the right).

Limit Exists If...

The limit exists only if left and right limits are equal:

Worked Examples

Example 1: Direct Substitution (Basic)

Find

For continuous functions, substitute directly:

Limit = 7

Example 2: Limit at a Hole (Intermediate)

Find

Direct substitution gives 0/0 (indeterminate). Factor:

Now substitute:

Limit = 4

Example 3: One-Sided Limits (Advanced)

For , find

Left:

Right:

Limit DNE (left ≠ right)

Common Pitfalls

Confusing limit with function value

The limit can exist even if f(c) is undefined or different!

Assuming 0/0 means the limit is 0

0/0 is indeterminate—you must simplify first.

Ignoring one-sided limits

Always check both sides when there's a discontinuity.

Real-World Application

Instantaneous Velocity

Your speedometer shows instantaneous speed—the limit of average velocity as the time interval approaches zero. This is exactly how derivatives work!