Not Every Forbidden Input Earns an Asymptote

You'll learnto decide when a forbidden input produces a vertical asymptote.

A vertical asymptote is the graph losing its composure: outputs running off without bound as the input closes in on a value the function cannot accept. But a shared factor changes everything, and multiplicity casts the deciding vote. Learn the test, say the behaviour in one-sided limits, and take inventory in a single pass.

8 STEPS · 6 QUICK CHECKS · MULTIPLICITY TEST · ONE-SIDED LIMITS · FULL INVENTORY · v2

denominator wins → asymptote · numerator holds → hole

Before you start

What you're looking at

A table of outputs for 1/(x−2) as the input closes in on 2 from both sides.

The question

As the input approaches the forbidden value, do the outputs run off without bound?

Watch for

The output magnitudes exploding, then the one-sided limit language that names the behavior.

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