Chapter 12.2, Problem 45E

To determine

To show: that the limit of $f$ as $t\to 2$ does not exist.

Explanation of Solution

Given information:

A current salary of $30000, the salaries $S$ for the next 3 years are given by

  $S\left(t\right)=\{\begin{array}{l}300000<t\le 1\\ 324001<t\le 2\\ 349922<t\le 3\end{array}$

Where $t$ represents the time in years.

Calculation:

Consider the following function that depicts the salary guaranteed by a union

  $S\left(t\right)=\{\begin{array}{l}300000<t\le 1\\ 324001<t\le 2\\ 349922<t\le 3\end{array}$

To check the existence of limit as $t$ approaches 2, proceed as follows.

Find the value of the function as $t$ approaches 2 from left and right.

When

  $\begin{array}{l}t=1.9\\ f\left(t\right)=32400\end{array}$

When

  $\begin{array}{l}t=1.99\\ f\left(t\right)=32400\end{array}$

When

  $\begin{array}{l}t=1.999\\ f\left(t\right)=32400\end{array}$

When

  $\begin{array}{l}t=2\\ f\left(t\right)=32400\end{array}$

When

  $\begin{array}{l}t=2.001\\ f\left(t\right)=33.992\end{array}$

When

  $\begin{array}{l}t=2.01\\ f\left(t\right)=33.992\end{array}$

When

  $\begin{array}{l}t=2.1\\ f\left(t\right)=33.992\end{array}$

As the value of the function approaches two different value as $t$ approaches 2 from left and right, the limit of the function does not exist at $t=2$ .