Lesson PreviewLesson 7 of 16 in this unit

1.7 Selecting Procedures for Determining Limits

Part of Limits and Continuity in AP® Calculus AB. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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A major calculus skill is choosing the correct method for a given limit problem. Some limits can be solved immediately by substitution, some require algebraic properties, some require factoring, and some require rationalizing. Good problem-solving means recognizing the structure of the expression before doing extra work. In general, you should first try direct substitution. If it works, you are done. If it gives 0/0, then you need a different method.

This lesson is about strategy rather than a new formula. Rational functions often suggest factoring if substitution fails. Radical expressions often suggest rationalizing. Piecewise functions, graphs, and tables usually require comparing left-hand and right-hand behavior. Learning to choose the simplest correct procedure saves time and reduces mistakes, especially on AP-style questions where efficiency matters.

Example 1: Find

\lim_{x \to 0}\frac{\sin x}{x}

This is a good example for Lesson 1.7 because direct substitution gives

\frac{0}{0}

but this is not a factoring or rationalizing problem like the ones in Lesson 1.6. Instead, this is a special trigonometric limit that must be recognized as its own standard result in calculus. So the correct procedure here is to identify the form rather than try ordinary algebraic manipulation.

Using the standard limit result,

\lim_{x \to 0}\frac{\sin x}{x}=1

So,

\lim_{x \to 0}\frac{\sin x}{x}=1

Make sure to memorize this limit (at least for now)!

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