Lesson 3.13

Vertical Asymptotes

A vertical asymptote is an invisible wall the graph approaches but never touches. It occurs where the denominator is zero (and the numerator isn't).

Introduction

Vertical asymptotes represent infinite discontinuities — places where the function shoots off toward . They occur at domain restrictions that don't cancel with the numerator.

Past Knowledge

Domain restrictions (Lesson 3.1) and the parent function (Lesson 3.12).

Today's Goal

Find vertical asymptotes by factoring and identifying non-cancelling zeros of the denominator.

Future Success

Lesson 3.14 covers what happens when a zero does cancel with the numerator (holes).

Key Concepts

Finding Vertical Asymptotes

1

Factor numerator and denominator completely

2

Cancel any common factors (those become holes — Lesson 3.14)

3

Set the remaining denominator factors = 0

4

Each solution is a vertical asymptote

Key Distinction

Vertical Asymptote

Denominator zero
Numerator ≠ zero

Hole

Both zero
(factor cancels)

Worked Examples

Example 1: Single VA

Basic

Find the vertical asymptote(s) of .

1

Set denominator = 0

VA:

Example 2: Two VAs

Intermediate

Find the vertical asymptote(s) of .

1

Factor denominator

Numerator is just — no common factors to cancel.

VAs: and

Example 3: VA vs. Hole

Advanced

Find the vertical asymptote(s) of .

1

cancels → that's a hole at

2

Remaining denominator:

VA: only (not — that's a hole)

Common Pitfalls

Calling Every Restriction a VA

You must check for cancellation first. If a factor cancels, it's a hole, not a VA.

Forgetting to Factor

If you skip factoring, you won't detect shared factors between the numerator and denominator.

Real-Life Applications

In physics, vertical asymptotes model situations where a quantity becomes infinitely large — like the electric field at a point charge, or the gravitational force as two objects approach zero separation distance.

Practice Quiz

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