Chapter 1.2, Problem 18E

To determine

To calculate: The limit of the expression $\underset{x\to 0^{-}}{\lim }\frac{\mathrm{int}x}{x}$ .

Answer to Problem 18E

The obtained value is 1.

Explanation of Solution

Given Information:

The given expression is $\underset{x\to 0^{-}}{\lim }\frac{\mathrm{int}x}{x}$ .

Calculation:

Consider the given expression,

As the given expression $\frac{\mathrm{int}x}{x}$ is showing the greatest integer function.

Now, draw the graph of the greatest integer function by using graphing calculator.

  

So the greatest integer function value for limit $0^{-}$ is $-1$ from the above graph.

  $\begin{array}{c}\underset{x\to 0^{-}}{\lim }\frac{\mathrm{int}x}{x}=\frac{\underset{x\to 0^{-}}{\lim }\left(\mathrm{int}x\right)}{\underset{x\to 0^{-}}{\lim }x}\\ =\underset{h\to 0}{\lim }\frac{-1}{0+h}\\ =\underset{h\to 0}{\lim }\frac{-1}{0}\\ =\infty \end{array}$

Therefore, the obtained limit value is $\infty$ .