Chapter 12, Problem 8PS

To determine

To find: $\underset{x\to 0}{\lim }f\left(x\right)$ and $\underset{x\to 0}{\lim }g\left(x\right)$ .

Answer to Problem 8PS

The limit of the function $\underset{x\to 0}{\lim }g\left(x\right)=0$ .

Explanation of Solution

Given information:

The given functions

  $f\left(x\right)=\{\begin{array}{l}0,\text{When}x\text{is}\text{rational}\\ 1,\text{When}x\text{is}\text{irrational}\end{array}$

And

  $g\left(x\right)=\{\begin{array}{l}0,\text{When}x\text{is}\text{rational}\\ x,\text{When}x\text{is}\text{irrational}\end{array}$

Calculation:

Find whether the values of the function as $x$ approaches 0 from left and right are same.

  $\underset{x\to 0}{\lim }f\left(x\right)$

As $x$ approaches from left or right, there are infinite numbers that are either rational or irrational numbers that are either rational or irrational numbers. Hence, the value of the function oscillates from 0 to 1. As the value of the function does not approach to a constant value, the limit of the function is not defined.

Consider the limit of function $\underset{x\to 0}{\lim }g\left(x\right)$

As the variable approaches from left or right, it has to approach from either a rational or irrational number.

Case (1)

When it approaches from rational

  $\underset{x\to 0^{-}}{\lim }g\left(x\right)=0$

Case (2)

When it approaches from irrational

  $\begin{array}{c}\underset{x\to 0^{-}}{\lim }g\left(x\right)=\underset{x\to 0^{-}}{\lim }x\\ =0\end{array}$

As 0 is a rational number, the value of function at $x=0$ is also 0.

Since the value of the function as the variable approaches 0 from either side or the value of the function at the point $x=0$ are same, the limit is defined.

Thus the limit of the function $\underset{x\to 0}{\lim }g\left(x\right)=0$ .