The limit of $x$ square minus four divided by $x$ minus $2$ is undefined as the value of $x$ approaches two when we try to evaluate the limit by the direct substitution method.

$\displaystyle \large \lim_{x\,\to\,2}{\normalsize \dfrac{x^2-4}{x-2}}$ $\,=\,$ $\dfrac{0}{0}$

The expression in the numerator $x$ squared minus four is a second degree polynomial and it can be factored. So, let us try to find the limit of square of $x$ minus $4$ divided by $x$ minus $2$ by the factoring method as the value $x$ closer to $2$.

The first term in the numerator is in square form. The input of limit operation tends to $2$ and the second term in the denominator of rational function is also $2$. So, it is recommended to express the second term in the numerator in terms of $2$ and also in square form.

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,2}{\normalsize \dfrac{x^2-2 \times 2}{x-2}}$

The number $2$ is multiplied by itself. So, let’s express the product of the factors in exponential notation.

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,2}{\normalsize \dfrac{x^2-2^2}{x-2}}$

Now, the difference of two squares can be factored by the difference of squares rule.

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,2}{\normalsize \dfrac{(x+2)(x-2)}{x-2}}$

The algebraic expression in the numerator is simplified as a product of two factors and there is nothing to simplify in the denominator. So, it is time to concentrate on simplifying the rational function.

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,2}{\normalsize \dfrac{(x+2) \times (x-2)}{x-2}}$

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,2}{\normalsize \dfrac{(x+2) \times \cancel{(x-2)}}{\cancel{x-2}}}$

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,2}{\normalsize (x+2)}$

The rational function $x$ square minus $4$ divided by $x$ minus $2$ is simplified as an algebraic expression $x$ plus $2$, and the simplified algebraic expression cannot be simplified further. So, the limit of the function can be now evaluated by the direct substitution method.

$=\,\,$ $2+2$

$=\,\,$ $4$

Latest Math Topics

Aug 31, 2024

Aug 07, 2024

Jul 24, 2024

Dec 13, 2023

Latest Math Problems

Oct 22, 2024

Oct 17, 2024

Sep 04, 2024

Jan 30, 2024

Oct 15, 2023

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved