Limits: by Conjugates
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Introduction

    This lesson page will inform you how to find limits of certain rational functions that contain square roots. The method that will be taught for this situation will be a method by conjugates. Here are the sections within this page:

    If your knowledge of conjugates and/or ability to reduce rational expressions is weak, review this material before proceeding.

    ideo: Factoring a Difference of Two Squares
    esson: Using Complex Conjugates for Dividing Complex Numbers



    Here is a complicated limit.

limit rational expression square root

    We can try substituting the value 5 into the expression, like so.

limit rational expression square root substitution zero denominator

    This result is a huge issue because it is illogical to have a denominator equal to zero. See this MATHguide video to learn why this is so.

    ideo: Division by Zero

    If we get a zero in the denominator, it means the result is indeterminant. It means we know nothing about the limit and that we have to try a different strategy. So, we are going to look at conjugates.

    The denominator of the original problem is this.

expression square root

    It has a conjugate, which is this.

conjugate expression square root

    We are going to multiply the numerator and the denominator of the original rational expression by this conjugate. Why? Doing the algebra will reveal how it will simplify the problem.

multiplying by conjugate rational expression square root

    There is a great deal of algebra to do, like doing the double distributive property when we multiply the denominators.

algebra analytical limit

    Here is another example.

limit rational expression square root

    The conjugate of the denominator is

limit rational expression square root conjugate

    We will multiply the numerator and the denominator by this conjugate, like so.

limit rational expression square root

    The remainder of the problem requires a number of algebraic steps, shown below.

limit rational expression square root steps

    ideo: Limits: By Conjugates
    uiz: Limits: By Conjugates


    Try this instructional video to learn this lesson.

    ideo: Limits: By Conjugates


    Try this interactive quizmaster to determine if you understand the lesson.

    uiz: Limits: By Conjugates


    Try these lessons, which are related to the sections above.

    esson: Limits of the Extreme
    esson: Limits of Ratios of Polynomials
    esson: Limits of Piecewise Functions
    esson: Limits by Numerical Approach
    esson: Limits by Factoring
    esson: Continuity