Solvequill Blog · math · 4 min read · 2 views

Limits: when substitution works and when it lies

A short guide to deciding whether to plug in the value, factor first, or look for a hidden cancellation.

Published: May 3, 2026

The first move in a limit should be cheap: substitute the target value. What happens next tells you the real method.

The main idea

If substitution gives a normal number, you are done. If it gives $\frac{0}{0}$ , factor or simplify because the expression is hiding a removable issue.

A short example

For $x \to 2 lim \frac{x ^{2} - 4}{x - 2}$ , substitution gives $\frac{0}{0}$ , so factor $x^{2} - 4 = (x - 2) (x + 2)$ .

x \to 2 lim \frac{x ^{2} - 4}{x - 2} = x \to 2 lim (x + 2) = 4

Check while you solve

Substitute once.
Treat $\frac{0}{0}$ as a signal to rewrite.
After canceling, substitute again.

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