Chapter 12, Problem 7PS

To determine

Find (if possible) $\underset{x\to 0}{\lim }f(x)$ and $\underset{x\to 0}{\lim g(x)}$ .

Answer to Problem 7PS

The limit of the function is not defined.

  $\boxed{\underset{x\to 0}{\lim g(x)}=0}$

Explanation of Solution

Given information:

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

And

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

Find (if possible) $\underset{x\to 0}{\lim }f(x)$ and $\underset{x\to 0}{\lim g(x)}$ . Explain your reasoning.

Calculation:

Consider the following functions

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

And

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

To determine the limit of the function as $x$ approaches $0$ , proceed as follows.

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

  $\underset{x\to 0}{\lim }f(x)$

As $x$ approaches from left or right, there are infinite 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,

Hence, the limit of the function is not defined.

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

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

When it approaches from rational

  $\underset{x\to 0}{\lim g(x)}=0$

When it approaches from irrational

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

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

Hence 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.

Hence, the limit of the function $\boxed{\underset{x\to 0}{\lim g(x)}=0}$