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AP®MathsCollege BoardCalculus ABStudy GuidesUnit 1: Limits & ContinuityLimitsEvaluating Limits Numerically & Graphically

Evaluating Limits Numerically & Graphically (College Board AP® Calculus AB)

Study Guide

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Limits from tables

How can I estimate a limit using values in a table?

  • Values of a function in a table can show the behavior of a function near a point

    • This can allow you to estimate the limit at that point

  • For example, let f be the function defined by f open parentheses x close parentheses equals fraction numerator 1 minus cos x over denominator x squared end fraction

    • The table below shows values of the function near x equals 0

    • Note that the function is not defined at x equals 0, because f open parentheses 0 close parentheses equals 0 over 0

x

f(x)

-0.1

0.49958347

-0.01

0.49999583

-0.001

0.49999996

0

not defined

0.001

0.49999996

0.01

0.49999583

0.1

0.49958347

  • From the table we can see that f open parentheses x close parentheses gets nearer and nearer to 0.5 as x gets nearer and nearer to 0

    • Therefore we can estimate that limit as x rightwards arrow 0 of f open parentheses x close parentheses is equal to 0.5

    • Analytical methods would need to be used to confirm that this is indeed the limit

Limits from graphs

How can I estimate a limit using a graph?

  • A graph can show the behavior of a function near a point

    • This can allow you to estimate the limit at that point

  • For example, let f be the function defined by f open parentheses x close parentheses equals fraction numerator 1 minus cos x over denominator x squared end fraction

    • The graph below shows the behavior the function near x equals 0

    • Note that the function is not defined at x equals 0, because f open parentheses 0 close parentheses equals 0 over 0

A graph of the function y=(1-cosx)/x^2 between x=-2 and x=2, showing limiting behavior as x approaches zero

  • From the graph we can see that f open parentheses x close parentheses gets nearer and nearer to 0.5 as x gets nearer and nearer to 0

    • Therefore we can estimate that limit as x rightwards arrow 0 of f open parentheses x close parentheses is equal to 0.5

    • Analytical methods would need to be used to confirm that this is indeed the limit

Examiner Tips and Tricks

You can graph functions on your graphing calculator to check your answers when determining limits analytically.

Horizontal asymptote

What is a horizontal asymptote?

  • A horizontal asymptote is a horizontal line

    • that the graph of a function gets closer and closer to (but never touches or intersects)

    • as x becomes unbounded in the positive or negative direction

  • On the following diagram, the horizontal asymptote is indicated by a dashed line

A graph of a function f with a horizontal asymptote at y=c
An example of a function with a horizontal asymptote at y=c

How can I identify horizontal asymptotes using limits?

  • A function will have a horizontal asymptote if it has a finite limit at infinity

    • I.e. the line y equals c will be a horizontal asymptote for the graph of a function f if

      • limit as x rightwards arrow infinity of f open parentheses x close parentheses equals c, or

      • limit as x rightwards arrow negative infinity of f open parentheses x close parentheses equals c

  • Horizontal asymptotes (if any) may therefore be determined by evaluating the limits at infinity

Examiner Tips and Tricks

By graphing a function on your graphing calculator you can:

  • spot any asymptotic behavior by a function at plus or minus infinity

  • check limits that you have determined analytically

Vertical asymptote

What is a vertical asymptote?

  • A vertical asymptote is a vertical line

    • that the graph of a function gets closer and closer to (but never touches or intersects)

    • as x gets closer and closer to the x-value of the vertical line

  • On the following diagram, the vertical asymptote is indicated by a dashed line

Graph of a function f with a vertical asymptote at x=c
An example of a function with a vertical asymptote at x=c

How can I identify vertical asymptotes using limits?

  • A function will have a vertical asymptote at any x-value where the function becomes unbounded

    • I.e. the line x equals c will be a vertical asymptote for the graph of a function f if

      • limit as x rightwards arrow c to the power of minus of f open parentheses x close parentheses equals plus-or-minus infinity, or

      • limit as x rightwards arrow c to the power of plus of f open parentheses x close parentheses equals plus-or-minus infinity

  • Vertical asymptotes (if any) may therefore be determined by identifying points where the function becomes unbounded

    • Usually this will involve a function in the form of a quotient

      • at points where the denominator becomes zero

Examiner Tips and Tricks

By graphing a function on your graphing calculator you can:

  • spot any asymptotic (i.e. unbounded) behavior by a function at certain values of x

  • check that vertical asymptotes you determine analytically are correct

Worked Example

Let f be the function defined by f open parentheses x close parentheses equals fraction numerator 3 x minus 11 over denominator x minus 2 end fraction.

Using limits, identify the vertical and horizontal asymptotes (if any) on the graph of f.

Answer:

The denominator becomes 0 when x equals 2, so start by considering the limits there

At 2 the numerator is equal to -5, so zero only occurs in the denominator

Just 'to the left' of 2, 3 x minus 11 less than 0 and x minus 2 less than 0 so

limit as x rightwards arrow 2 to the power of minus of f open parentheses x close parentheses equals infinity

Just 'to the right' of 2, 3 x minus 11 less than 0 and x minus 2 greater than 0 so

limit as x rightwards arrow 2 to the power of plus of f open parentheses x close parentheses equals negative infinity

This confirms that the graph of f has a vertical asymptote at x equals 2

To identify horizontal asymptotes, start by rearranging to make the behavior of the function more obvious

fraction numerator 3 x minus 11 over denominator x minus 2 end fraction equals fraction numerator 3 open parentheses x minus 2 close parentheses minus 5 over denominator x minus 2 end fraction equals fraction numerator 3 open parentheses x minus 2 close parentheses over denominator x minus 2 end fraction minus fraction numerator 5 over denominator x minus 2 end fraction equals 3 minus fraction numerator 5 over denominator x minus 2 end fraction

fraction numerator 5 over denominator x minus 2 end fraction becomes closer and closer to zero as x increases in the positive or negative directions, so

limit as x rightwards arrow negative infinity of f open parentheses x close parentheses equals limit as x rightwards arrow infinity of f open parentheses x close parentheses equals 3 minus 0 equals 3

This means that the graph of f has a horizontal asymptote at y equals 3

The graph of f has a vertical asymptote at x equals 2, and a vertical asymptote at y equals 3

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    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.