Calculus I · Limits and Continuity · review

Limits and Continuity Unit Review

Mastery Review and Practice Exams

The Complete Limit Decision Process

A strong student does not merely know many techniques. A strong student can decide which technique applies when the problem is not labeled.

Method

Master Decision Tree

When evaluating a limit:

• Read the direction. Is the input approaching a finite number, one side of a finite number, $+\infty$ , or $-\infty$ ? • Substitute when meaningful.

• Real number: usually finished. • $0/0$ : algebraic indeterminate form. • Nonzero over zero: one-sided sign analysis. • $\infty/\infty$ : compare powers or divide by the dominant power. • $\infty-\infty$ : combine or rationalize.

• Inspect the structure.

• polynomial factors $\to$ factor and cancel; • radicals $\to$ conjugate; • fractions inside fractions $\to$ common denominator; • absolute values/piecewise rules $\to$ one-sided cases; • bounded oscillation times zero $\to$ squeeze; • trig near zero $\to$ standard trig limits; • rational function at infinity $\to$ degrees/dominant terms; • radical at negative infinity $\to$ use $|x|$ .

• State the conclusion precisely. Distinguish a finite number, $\mathrm{DNE}$ , $+\infty$ , and $-\infty$ .

After the explanation

Use the section idea

Reading lens

Can you diagnose the limit type and justify a method before beginning the algebra?

Mental model

A mixed problem is a classification task before it is a calculation: direction, substitution result, structure, and required conclusion determine the route.

Decision

Name the limit type and first legal move in a margin note, then solve and check whether the conclusion matches the graph or sign behavior.

Common trap

Pattern matching without diagnosis makes similar-looking problems blur together and hides whether the error was conceptual, algebraic, or strategic.

Check yourself

You are exam-ready when you can choose a method without a section label, explain the choice, and correct a miss by naming its exact cause.

Source & rights

Original instruction with traceable references.

The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.

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