Limits: by Factoring
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Search | Updated July 28th, 2023

    This lesson page will inform you how to find limits of certain rational functions using polynomial factors. Here are the sections within this page:

    If your factoring and reducing of rational expression skills need assistance, review these lessons before proceeding.

    esson: Factoring
    esson: Rational Expressions

    For some limits, the problem can be solved using substitution. Here is an example of such a limit.

finding limit via substitution

    We need only replace the xís with 0, like so.

finding limit via substitution

    Here are two more examples.

finding limit via substitution

    However, not all problems are so simple. The next section will show how substitution alone is not always a reliable technique for calculating a limit.

    In the last section, we saw how we could use substitution for this problem.

finding limit via substitution

    Now, letís see what happens if we use substitution as we approach x = 3 instead of x = 0.

finding limit via substitution zero denominator

    This is a call for mathematical alarm because the denominator cannot be equal to zero. Why? This video explains why division by zero is illogical.     Anytime we substitute a value into a function and get a denominator equal to zero, we get an undetermined result. It means the process tells us absolutely nothing about the limit. It also means we have to try some other strategy.

    The other strategy is to factor the polynomial.

    If we factor the function, we can reduce the rational function, like so.

finding limit via factoring

    Now we can use substitution because we no longer get a zero in the denominator when we do so.

limit substitution

    Here are two more examples using factoring as a strategy.

finding limit via factoring two examples

    ideo: Limits: By Factoring
    uiz: Limits: By Factoring

    Try this instructional video to learn this lesson.

    ideo: Limits: By Factoring

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

    uiz: Limits: By Factoring

    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 Conjugates
    esson: Continuity
    esson: Sine Coefficient Limit Problem