Selecting Procedures for Determining Limits (College Board AP® Calculus AB)

Selecting procedures for determining limits

How do I choose the correct procedure for evaluating a limit?

• You should be familiar with all the different methods for estimating or evaluating limits

  • This way you can choose the most appropriate method to use for an exam question

• You can estimate the value of a limit using

  • A table of values for a function

  • The graph of a function

    • The graph may be given in the question, or produced on your graphing calculator

  • See the 'Evaluating Limits Numerically & Graphically' study guide

  • An estimate is not enough to absolutely determine the value of a limit

    • But it can be used to check limits that you determine analytically

• You can evaluate a limit analytically using

  • Substitution

    • For a function continuous on an interval including the point in question

    • For limits at the 'joins' of piecewise functions

  • Simplifying algebraically

    • For quotient functions where the numerator and/or denominator can be factorised and common factors canceled

  • Multiplying by a conjugate

    • For quotient functions involving surds, e.g.

  • Multiplying by a reciprocal

    • For quotient functions involving polynomials, e.g.

  • See the 'Evaluating Limits Analytically' study guide

• You can also evaluate a limit using

  • The squeeze theorem

    • For a function bounded above and below by two other functions whose limits are equal at a point

  • Trigonometric limit theorems

    • Using the results and along with algebraic manipulation and properties of limits

  • See the 'Squeeze Theorem & Trigonometric Limits' study guide

Are there any hidden tricks for evaluating limits?

• Remember that certain processes in calculus are defined using limits

  • If you recognise a limit as matching one of these definitions, you may be able to use the process to find the value of the limit

• The derivative of a function is defined as (see the 'Derivatives & Tangents' study guide)

  • Consider

    • so this is equivalent to

    • That's the definition of the derivative of at (just using instead of )

    • 

    • So

• The definite integral of a function is defined as the limit of Riemann sums (see the 'Properties of Definite Integrals' study guide)

  • Consider over the interval between and

    • That is just the definition of

    • And

    • So over the interval between and ,