Calculus I · Limits and Continuity · review
Finite Limits Review
Section 2 Summary
• Direct substitution is the correct first move, not an unsophisticated shortcut. • A legal real substitution result usually finishes the problem. • is indeterminate; it tells you to transform the expression. • Factor polynomial forms, rationalize radicals, combine complex fractions, and use one-sided rules for absolute values. • Cancellation is legal in a limit when it is performed for nearby non-target inputs where the cancelled factor is nonzero.
After the explanation
Use the section idea
What did direct substitution reveal, and which algebraic move removes the obstacle without changing nearby behavior?
Substitution is a diagnostic first move: a real number usually finishes the problem, while an indeterminate form asks for a structural rewrite.
Match the obstacle to the algebra—factor polynomial zeros, rationalize radicals, combine complex fractions, and split absolute values into one-sided cases.
Zero over zero is not an answer, and cancellation is legal only for factors after the expression has been rewritten as a product.
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.
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