Chapter 1.1, Problem 38E

To determine

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

Answer to Problem 38E

  $\underset{x\to 0^{-}}{\lim }\mathrm{int}x=-1$

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 $x=-0.9999$ , $y=-1$ and if $x=-0.99$ , $y=-1$ ,

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 $0\ge x$ ,

  $\underset{x\to 0^{-}}{\lim }\mathrm{int}x=-1$

Hence, value of the limit is $-1$ .