Limits: when substitution works and when it lies

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

Use this post as a small study note for making better decisions while solving, not as a quick definition dump. Read the idea, pause at the example and try to move one line forward yourself, then use the checklist on a real problem.

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.

The important part is not memorizing the method; it is noticing what calls the method into action. Which expression, diagram, unit, variable, or code behavior made this approach the right one? If you can name that trigger, a similar problem becomes much less stressful.

A short example

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

How to approach it step by step

1. Write the givens cleanly and fix units, symbols, or variable names before calculating.
2. Identify the intermediate fact you need before trying to jump directly to the answer.
3. After each line, run a small check: do the units match, is the sign correct, is indentation or scope correct in code?
4. When you get an answer, return to the original question and check that it answers exactly what was asked.

Check while you solve

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

The common mistake

Cancel factors, not terms. That one sentence prevents many invalid limit solutions.

The practical way to catch this mistake is to check the decision points, not only the final answer. Ask questions like: Why this formula? Why this component? Why this loop condition? That makes the answer stronger in content, not just in arithmetic.