PreCalculus Continuity, End Behavior, and Limits

By Jeff Twiddy

Pre-calculus doesn’t have to be intimidating! In this lesson, you’ll discover how continuity, end behaviors, and limits interconnect through intuitive “if this, then that” scenarios that make learning fun and interactive.

Unlocking the Mystery Behind Functions

Understanding the behavior of functions can be both insightful and exciting. By exploring continuity, end behavior, and limits, students learn how functions act at their extremes and what happens as they approach particular values—even if they never quite get there. This lesson demystifies these key concepts while providing plenty of hands-on opportunities to engage with the material.

What Is Continuity?

Continuous Functions

Continuous functions are smooth and unbroken. They have:

No Breaks or Holes: The graph can be traced with a pencil without ever lifting it.
Smooth Transitions: There are no sudden jumps or gaps in the graph.

Discontinuous Functions

In contrast, discontinuous functions are those that break the smooth flow:

Definition: Functions that are not continuous.
Types of Discontinuities:
- Infinite Discontinuity: Often shown as asymptotes where the function heads toward infinity.
- Jump Discontinuity: The graph abruptly “jumps” to a different value.
- Removable Discontinuity: Represented by an open circle in the graph, indicating a “hole.”

Exploring End Behavior

Understanding End Behavior

End behavior describes how a function behaves as $x$ increases or decreases without bound:

Arrow Direction: The direction of the arrows on the ends of the graph tells you if the function rises or falls.
Degree of the Function: For example, if the degree of the function is even, both ends of the graph will point in the same direction.

Real-World Analogy

Think of end behavior like watching a road that stretches into the horizon. As you drive further, you can predict whether the road will level off, keep climbing, or descend—even if you never see the very end.

Limits: Approaching Without Touching

What Are Limits?

Limits describe where a function is headed as it approaches a particular value:

Concept: It’s not necessary for the function to ever actually reach that value.
Classroom Activity: Imagine a student walking halfway to a wall repeatedly. With each step, they get closer, but theoretically, they never actually reach the wall. This “approaching” behavior perfectly illustrates the concept of limits.

An Engaging Introduction

For an exciting introduction to limits, check out this GREAT INTRO LESSON ON LIMITS that uses interactive activities to bring the concept to life.

Continuity, End Behavior, and Limits Lesson Materials & Downloadable Resources

Enhance your classroom experience with these ready-to-use, free materials:

1-3 Assignment – Continuity, End Behavior, and Limits

1-3 Bell Work – Continuity, End Behavior, and Limits

1-3 Exit Quiz – Continuity, End Behavior, and Limits

1-3 Guided Notes SE – Continuity, End Behavior, and Limits

1-3 Slide Show – Continuity, End Behavior, and Limits

Additional resources available for members:

1-3 Guided Notes Teacher Edition (Members Only)

1-3 Lesson Plan (Members Only)

1-3 Online Activities (Members Only)

1-3 Video Lesson (Members Only)

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