Question

How do you find the limit of $\dfrac{{|x|}}{x}$ as $x$ approaches 0?

Answer 1

Hint: We will first notice that \[|x| = \left\{ {\begin{array}{*{20}{c}}
  {x,x > 0} \\
  { - x,x < 0}
\end{array}} \right.\]. Then, we will put in both the possibilities, then we will get different limit for both and thus the limit of the function $\dfrac{{|x|}}{x}$ does not exist.

Complete step by step solution:
We are given that we are required to find the limit of $\dfrac{{|x|}}{x}$ as $x$ approaches 0.
Since we have to make x approach to 0.
We know that 0 can be approached from the left side of the number line and the right side of the number line.
Now, since we also know that the modulus of a function is defined as follows:-
$ \Rightarrow |x| = \left\{ {\begin{array}{*{20}{c}}
  {x,x > 0} \\
  { - x,x < 0}
\end{array}} \right.$
Therefore, when x approaches from the left side of the number line, we have the following expression with us:-
$ \Rightarrow \mathop {\lim }\limits_{x \to {0^ - }} \dfrac{{|x|}}{x} = \mathop {\lim }\limits_{x \to {0^ - }} \dfrac{{ - x}}{x}$
Simplifying the right hand side by crossing – off x from both the numerator and denominator, we will then obtain the following expression with us:-
$ \Rightarrow \mathop {\lim }\limits_{x \to {0^ - }} \dfrac{{|x|}}{x} = - 1$
Now, when x approaches from the right side of the number line, we have the following expression with us:-
$ \Rightarrow \mathop {\lim }\limits_{x \to {0^ + }} \dfrac{{|x|}}{x} = \mathop {\lim }\limits_{x \to {0^ + }} \dfrac{x}{x}$
Simplifying the right hand side by crossing – off x from both the numerator and denominator, we will then obtain the following expression with us:-
$ \Rightarrow \mathop {\lim }\limits_{x \to {0^ + }} \dfrac{{|x|}}{x} = 1$
Now, since the limits are different from both approaches, therefore, its limit does not exist.

Hence, the answer is limit of the given function $\dfrac{{|x|}}{x}$ when $x$ approaches 0 is non – existent.

Note: The students must note that we crossed – off x while finding both the limits, we could cross that off because x is not equal to 0. We can never cross – off or cancel any variable which has even the slightest possibility of being equal to 0.
The students must note that here, we are just working in one – dimensional real numbers, therefore, to approach 0, we only have two options that are left and right which is the negative side and the positive side of the number line respectively.