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AP®MathsCollege BoardCalculus ABStudy GuidesUnit 1: Limits & ContinuityLimitsProperties of Limits

Properties of Limits (College Board AP® Calculus AB)

Study Guide

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Properties of limits

What properties of limits do I need to know?

  • There are a number of limit properties (also known as limit theorems) that you need to know and be able to use

    • They let you work out the limits of complicated functions algebraically by combining the limits of simpler functions

  • The limit of a constant function: If k is a constant then

    • limit as x rightwards arrow a of k equals k

  • The limit of a multiple of a function: If k is a constant and limit as x rightwards arrow a of f open parentheses x close parentheses equals L, then

    • limit as x rightwards arrow a of open parentheses k f open parentheses x close parentheses close parentheses equals k L

  • The limit of a sum or difference of functions: If limit as x rightwards arrow a of f open parentheses x close parentheses equals L and limit as x rightwards arrow a of g open parentheses x close parentheses equals M, then

    • limit as x rightwards arrow a of open parentheses f open parentheses x close parentheses plus-or-minus g open parentheses x close parentheses close parentheses equals L plus-or-minus M

  • The limit of a product of functions: If limit as x rightwards arrow a of f open parentheses x close parentheses equals L and limit as x rightwards arrow a of g open parentheses x close parentheses equals M, then

    • limit as x rightwards arrow a of open parentheses f open parentheses x close parentheses times g open parentheses x close parentheses close parentheses equals L times M

  • The limit of a quotient of functions: If limit as x rightwards arrow a of f open parentheses x close parentheses equals L and limit as x rightwards arrow a of g open parentheses x close parentheses equals M with M not equal to 0, then

    • limit as x rightwards arrow a of fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction equals L over M

  • The limit of the power of a function: If limit as x rightwards arrow a of f open parentheses x close parentheses equals L and n is a real number, then

    • limit as x rightwards arrow a of open square brackets f open parentheses x close parentheses close square brackets to the power of n equals L to the power of n

  • The limit of a composite function: If limit as x rightwards arrow a of f open parentheses x close parentheses equals L and if the function g is continuous at x equals L, then

    • limit as x rightwards arrow a of g open parentheses f open parentheses x close parentheses close parentheses equals g open parentheses L close parentheses

  • Note that statements like limit as x rightwards arrow a of f open parentheses x close parentheses equals L and limit as x rightwards arrow a of g open parentheses x close parentheses equals M assume that those limits exist, and that L and M are real numbers

Examiner Tips and Tricks

Make sure that the necessary conditions are met before using one of the limit properties, for example:

  • M not equal to 0 for the quotient property

  • g is continuous at x equals L for the composite function property

  • Both limits, if being combined, are tending toward the same value open parentheses a close parentheses

Worked Example

Let f and g be functions such that limit as x rightwards arrow 3 of f open parentheses x close parentheses equals 7 and limit as x rightwards arrow 3 of g open parentheses x close parentheses equals negative 2.

Let h be a function that is continuous for all real numbers, and is such that h open parentheses negative 2 close parentheses equals 0 and h open parentheses 7 close parentheses equals 13.

Find the following limits:

(a) limit as x rightwards arrow 3 of open parentheses f open parentheses x close parentheses plus 4 close parentheses

Answer:

Note that limit as x rightwards arrow 3 of open parentheses 4 close parentheses equals 4, then use the limit of a sum of functions property

limit as x rightwards arrow 3 of open parentheses f open parentheses x close parentheses plus 4 close parentheses equals 7 plus 4

limit as x rightwards arrow 3 of open parentheses f open parentheses x close parentheses plus 4 close parentheses equals 11

(b) limit as x rightwards arrow 3 of open parentheses g open parentheses x close parentheses minus 2 f open parentheses x close parentheses close parentheses

Answer:

Use the limit of a multiple of a function property, along with the limit of a difference of functions property

limit as x rightwards arrow 3 of open parentheses g open parentheses x close parentheses minus 2 f open parentheses x close parentheses close parentheses equals negative 2 minus 2 open parentheses 7 close parentheses

limit as x rightwards arrow 3 of open parentheses g open parentheses x close parentheses minus 2 f open parentheses x close parentheses close parentheses equals negative 16

(c) limit as x rightwards arrow 3 of open parentheses fraction numerator g open parentheses x close parentheses over denominator f open parentheses x close parentheses end fraction close parentheses

Answer:

Use the limit of a quotient of functions property

limit as x rightwards arrow 3 of open parentheses fraction numerator g open parentheses x close parentheses over denominator f open parentheses x close parentheses end fraction close parentheses equals fraction numerator negative 2 over denominator 7 end fraction

limit as x rightwards arrow 3 of open parentheses fraction numerator g open parentheses x close parentheses over denominator f open parentheses x close parentheses end fraction close parentheses equals negative 2 over 7

(d) limit as x rightwards arrow 3 of h open parentheses f open parentheses x close parentheses close parentheses

Answer:

Use the limit of a composite function property

Note that we are told that h is continuous for all real numbers, so the property is valid for use here

limit as x rightwards arrow 3 of h open parentheses f open parentheses x close parentheses close parentheses equals h open parentheses 7 close parentheses

limit as x rightwards arrow 3 of h open parentheses f open parentheses x close parentheses close parentheses equals 13

How do the properties of limits work with infinite limits?

  • There are several properties of limits involving infinite limits

    • They help determine whether a function increases without bound (i.e. tends to infinity) or decreases without bound (i.e. tends to negative infinity) at a particular point

  • Limit of bold 1 over bold italic x to the power of n at zero: If n is a positive integer, then

    • limit as x rightwards arrow 0 to the power of plus of 1 over x to the power of n equals infinity

    • limit as x rightwards arrow 0 to the power of minus of 1 over x to the power of n open curly brackets table row cell infinity space space if space n space is space even end cell row cell negative infinity space space if space n space is space odd end cell end table close

  • Infinite limits of quotients: If limit as x rightwards arrow a of f open parentheses x close parentheses equals L and limit as x rightwards arrow a of g open parentheses x close parentheses equals 0, then

    • if L greater than 0, limit as x rightwards arrow a of fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction equals open curly brackets table row cell infinity comma space space if space g open parentheses x close parentheses greater than 0 space as space x space approaches space a end cell row cell negative infinity comma space space if space g open parentheses x close parentheses less than 0 space as space x space approaches space a end cell end table close

    • if L less than 0, limit as x rightwards arrow a of fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction equals open curly brackets table row cell negative infinity comma space space if space g open parentheses x close parentheses greater than 0 space as space x space approaches space a end cell row cell infinity comma space space if space g open parentheses x close parentheses less than 0 space as space x space approaches space a end cell end table close

    • In both these cases, the limits from the left (as x rightwards arrow a to the power of minus) and right (as x rightwards arrow a to the power of plus) may be different

      • Check the behaviour of g open parentheses x close parentheses to determine the correct limit

Worked Example

Let f be a function such that limit as x rightwards arrow 0 of f open parentheses x close parentheses equals 1.

Let g be the function defined by g open parentheses x close parentheses equals x cubed.

Find limit as x rightwards arrow 0 to the power of minus of fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction and limit as x rightwards arrow 0 to the power of plus of fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction.

Answer:

We can use the infinite limits of quotient properties here

In both cases, limit as x rightwards arrow 0 of f open parentheses x close parentheses equals L greater than 0

For the limit from the left, note that g open parentheses x close parentheses equals x cubed is negative as it approaches 0 through the negative numbers

limit as x rightwards arrow 0 to the power of minus of fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction equals negative infinity

For the limit from the right, note that g open parentheses x close parentheses equals x cubed is positive as it approaches 0 through the positive numbers

limit as x rightwards arrow 0 to the power of plus of fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction equals infinity

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    Previous:Infinite Limits & Limits at InfinityNext:Evaluating Limits Analytically
    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.