Chapter 1.1, Problem 37E

To determine

To evaluate: the limit of the given function by substitution.

Answer to Problem 37E

  $\underset{x\to 0^{+}}{\lim }\mathrm{int}x=0$

Explanation of Solution

Given information:

The given limit is

  $\underset{x\to 0^{+}}{\lim }\mathrm{int}x$

Calculation:

  $y=\mathrm{int}x$ is the greatest integer function which means the output $y$ is the greatest integer that is less than or equal to $x$ . This means that whatever $x$ is, it gets rounded down to the nearest integer.

For example, if $x=0.1$ , $y=0$ and if $x=0.6$ , $y=0$ ,

The greatest integer function is then made up of horizontal line segments with jump discontinuities at each integer.

The function $\mathrm{int}x$ is nota function that provides the nearest integer to $x$ . It provides the greatest integer, that is smaller than $x$ this is also known as the floor function.

For $0\le x<1$ ,

  $\begin{array}{c}\underset{x\to 0^{+}}{\lim }\mathrm{int}x=\mathrm{int}0\\ =0\end{array}$

Hence, value of the limit is $0$ .