Linear Algebra For Dummies
Book image
Explore Book Buy On Amazon

When simply plugging the arrow number into a limit expression doesn't work, you can solve a limit problem using a range of algebraic techniques. These can include factoring, cancelling and conjugate multiplication.

Of course, before you try any algebra, your first step should always be to plug the arrow-number into the limit expression. If the function is continuous at the arrow-number (which it usually will be) and if plugging in results in an ordinary number, then that's the answer. You're done. For example, to evaluate

The limit of a function.

just plug in the arrow-number. You get

Replace the variable with the arrow-number.

That's all there is to it. Don't forget to plug in!

When plugging in fails because it gives you


you've got a nontrivial limit problem and a bit of work to do. You have to convert the fraction into some expression where plugging in does work. Here are some algebraic methods you can try:

  • FOILing

  • Factoring

  • Finding the least common denominator

  • Canceling

  • Simplification

  • Conjugate multiplication

Some of these methods are illustrated in the following examples.

Practice questions

  1. Solve the following limit:

    The limit of the function x-1/squared x + x - 2
  2. Solve the following limit:

    The limit of x - 9 divided by three minus square root of x

Answers and explanations

  1. The answer is 1/3.

    To obtain the answer, you need to factor, cancel, and plug in.

    The steps necessary to solve a limit.
  2. The answer is –6.

    This one is a bit more involved.

    You start by multiplying the numerator and denominator by the conjugate of the denominator,

    three plus the square root of x

    Now multiply out the part of the fraction containing the conjugate pair (the denominator in this problem).

    Multiplying the numerator and denominator of a function by the conjugate of the denominator.


    The limit of a function.

    Remember that any fraction of the form

    a-b divided by b-a.

    always equals –1.

    Now plug in.

    The solution for the limit of a function with a fraction.

About This Article

This article is from the book:

About the book author:

Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies.

This article can be found in the category: