Solving Limits with Algebra — Practice Questions

By Mark Ryan

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.