Calculus I · Limits and Continuity · reference

Finite Limit Decision Tree

The Finite-Limit Decision Tree

Method

A Repeatable Homework Procedure

For a finite target $x\to a$ :

• Substitute $x=a$ . • If you get a real number, stop. • If you get $0/0$ , inspect the expression:

• factor polynomial expressions; • rationalize radical differences; • combine complex fractions; • split absolute values or piecewise functions into one-sided cases; • save trig limits for Section 3.

• Simplify only for nearby values where the original expression is defined. • Substitute again. • State why the method is valid, especially if one-sided limits are involved.

After the explanation

Use the section idea

Reading lens

What did direct substitution reveal, and which algebraic move removes the obstacle without changing nearby behavior?

Mental model

Substitution is a diagnostic first move: a real number usually finishes the problem, while an indeterminate form asks for a structural rewrite.

Decision

Match the obstacle to the algebra—factor polynomial zeros, rationalize radicals, combine complex fractions, and split absolute values into one-sided cases.

Common trap

Zero over zero is not an answer, and cancellation is legal only for factors after the expression has been rewritten as a product.

Check yourself

You are ready to move on when you can justify why each rewrite preserves nearby values even if the original expression is undefined at the target.

Source & rights

Original instruction with traceable references.

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