Lesson 6.5

Removable Discontinuities

Also known as "Holes". When a problem disappears if you just simplify it enough.

Introduction

Prerequisite Connection: In Lesson 6.1, we canceled factors like and said the answer was 1. But we noted that still couldn't be 2.

Today's Increment: That forbidden point isn't a wall (Asymptote). It's just a missing pixel. It's a tiny, invisible hole in the line.

Why "Removable"? Because we can "fix" the function by simply plugging that one single hole with a definition. Asymptotes, on the other hand, are unfixable broken infinite gaps.

Explanation of Key Concepts

How to Find the Coordinates

Step 1: Factor and Cancel

If a factor cancels from Top AND Bottom, there is a HOLE at .

Step 2: Find the Y-value

Take the x-value of the hole and plug it into the SIMPLIFIED version of the equation.

If you plug it into the original, you define

(error).
If you plug it into the simplified, you get the location of the hole.

Worked Examples

Level: Basic

Example 1: Single Hole

Find the coordinates of the hole in .

Factor:

Cancel

.
Simplified Function:

(It's just a line!)

Find Coordinate

Set cancelled factor to 0:

Plug

into simplified:

Answer

Hole at

Level: Intermediate

Example 2: Distinguishing Features

Identify all discontinuities for .

CANCELS. Values that cancel are HOLES.
REMAINS. Values that stay in bottom are ASYMPTOTES.

Calculate Hole Y-value

Simplified:

Final Analysis

VA at

Hole at

Level: Advanced

Example 3: Multiple Removals

Analyze .

Factor Top:

BOTH

and

cancel out completely!
Simplified:

Hole 1 (from x):

. Plug into

. Point:

.
Hole 2 (from x-1):

. Plug into

. Point:

Common Pitfalls

Plugging x into the ORIGINAL function:
If you do this, you will get . Your calculator will say ERROR. That is correct—it is undefined there! We use the simplified version to find where the hole would be.
Forgetting the Hole entirely:
After you simplify to , it looks like a perfect line. You MUST remember to draw the little open circle at x=0. The simplification hides the history of the division by zero.

Real-World Application

Calculus: The Limit

The entire concept of a "Derivative" in Calculus describes the slope between two points as they get closer and closer together.

The formula ends up being at the exact instant of tangency. We use the concept of a Removable Discontinuity to "fill in" that hole and discover the Instantaneous Rate of Change. The hole IS the answer!