The limit of a rational function is indeterminate in a special case, when we try to find the limit by direct substitution. It is mainly due to a common factor in the expressions of both numerator and denominator. So, factoring polynomials in a rational function eliminates the indeterminate form, then the limit can be evaluated and this method is called evaluating the limits by factoring or factorisation.
Now, let’s learn more about finding the limit by factoring the polynomials in a rational function.
You should have complete knowledge on how to factorise polynomials. Then only, you can understand this method. So, learn the methods of factoring expressions firstly.
The methods of factorisation to learn how to factorize the polynomials in mathematics.
Once you know the six methods of factoring the polynomials, then you are ready to learn how to find the limits by factorization in calculus.
Let’s learn how to find the limit by factoring from a simple understandable limit problem.
Evaluate $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize \dfrac{x^2-4}{x(x-2)}}$
Firstly, let’s use the direct substitution method to find its limit.
$=\,\,$ $\dfrac{2^2-4}{2(2-2)}$
$=\,\,$ $\dfrac{2 \times 2-4}{2 \times (2-2)}$
$=\,\,$ $\dfrac{4-4}{2 \times 0}$
$=\,\,$ $\dfrac{0}{0}$
It expresses that the direct substitution method is failed to find the limit. There is a hidden common factor in both expressions. So, let’s use a method of factoring the expressions to eliminate the greatest common factor in them.
$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize \dfrac{x^2-2^2}{x(x-2)}}$
$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize \dfrac{(x+2)(x-2)}{x(x-2)}}$
$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize \dfrac{(x+2) \times (x-2)}{x \times (x-2)}}$
$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize \dfrac{(x+2) \times \cancel{(x-2)}}{x \times \cancel{(x-2)}}}$
$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize \dfrac{x+2}{x}}$
Now, use the direct substitution method to find the limit of function.
$=\,\,$ $\dfrac{2+2}{2}$
$=\,\,$ $\dfrac{4}{2}$
$=\,\,$ $2$
The limit by using a method of factorization is possible. So, it is called the method of finding a limit by factorisation.
The simple example helps you to understand how to find the limits by factoring polynomials. Now, let’s improve our knowledge on finding the limits by factoring method to exercise some problems.
The practice limits questions on finding the limits by factoring the polynomials and solutions with understandable steps.
A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.
Copyright © 2012 - 2025 Math Doubts, All Rights Reserved