Chapter 13, Problem 2RE

To determine

To find: the value of the limit and then confirm the result by using graphing calculator.

Answer to Problem 2RE

The value of limit is $\frac{1}{2}$.

Explanation of Solution

Given information:

The limit is $\underset{t\to -1}{\lim }\frac{t+1}{t^3-t}$.

Calculation: to find the limit value, first factorise the denominator and then cancel out common factors and then simplify,

  $\begin{array}{l}\underset{t\to -1}{\lim }\frac{t+1}{t^3-t}=\underset{t\to -1}{\lim }\frac{t+1}{t\left(t+1\right)\left(t-1\right)}\text{ [factorise]}\\ \underset{t\to -1}{\lim }\frac{t+1}{t^3-t}=\underset{t\to -1}{\lim }\frac{1}{t\left(t-1\right)}\text{ [cancel out common factor]}\\ \underset{t\to -1}{\lim }\frac{t+1}{t^3-t}=\frac{1}{-1\left(-1-1\right)}\text{ [put }t=-1]\\ \underset{t\to -1}{\lim }\frac{t+1}{t^3-t}=\frac{1}{2}\text{ [simplify]}\end{array}$

Thus, the limit value is $\frac{1}{2}$.

By using graphing calculator, the graph of function can be obtained as:

  

Form the above graph it can be observed that when the value of $t$ approaches $-1$ the function approach $\frac{1}{2}$.