Chapter 1.1, Problem 40E

To determine

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

Answer to Problem 40E

  $\underset{x\to 1.99}{\lim }\mathrm{int}x=1$

Explanation of Solution

Given information:

The given limit is

  $\underset{x\to 1.99}{\lim }\mathrm{int}x$

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 $x=1.99$ , $y=1$ and if $x=1.999$ , $y=1$ ,

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 $1\le x<2$ ,

  $\begin{array}{c}\underset{x\to 1.99}{\lim }\mathrm{int}x=\mathrm{int}1.99\\ =1\end{array}$

Hence, value of the limit is $1$ .