Use limit properties to determine the limits of functions
Use algebraic manipulations to determine the limits of functions
I can evaluate a limit using factoring techniques and locating “holes” in a graph.
I can evaluate a limit using trig identities.
I can evaluate a limit by rewriting a complex function as a combination of simpler functions.
Quick Lesson Plan
In this lesson, students explore various strategies for evaluating limits. Connecting domain restrictions with function behavior is one of the primary goals in this activity. Students will review the usefulness of factoring expressions to find limits as well as use basic trigonometric identities to rewrite limit expressions into a more usable form. Multiplying by a conjugate and examining tabular data on their calculator are also explored. Throughout the lesson, we emphasize that students know why and when a certain approach works best, and not just follow a rote procedure.
These lessons present a great opportunity to review many of the algebraic operations students studied in previous mathematics classes. And since there can never be too much review of trigonometry, students will need to produce (or at least repeat) the ratios for the six basic trig functions. The properties used for final evaluation of a limit are intuitive for most students; it is the algebra required to implement the properties that consumes most of student time and effort.
When common factors appear in the numerator and denominator of a fraction, we say “one out” to indicate that the terms simplify (cancel) to 1. When opposite terms are added, we say “zero out” to indicate the terms simplify (cancel) to zero. Saying “cancel” is ambiguous and doesn’t require students to understand why terms disappear from an expression!
Limit evaluation questions can appear on the multiple choice section of the AP Exam so efficient selection of the correct solving technique (see Topic 1.7) and then following through with correct algebra will save precious time.
Domain restrictions for piecewise functions can be challenging: emphasize that a specific x-value may belong to the domain of only one of the definitions. Because we are still early in our conversations about limits, some students may still believe that a function undefined at x = c cannot have a limit as x approaches c.