Question

Given that \(\mathop {{\rm{lim}}}\limits_{x \to \pi } {\rm{cs}}{{\rm{c}}^2}x = \infty \), illustrate Definition 6 by

finding values of \(\delta \) that correspond to (a) \(M = 500\) and (b) \(M = 1000\).

Short answer

(a) The value is \(\delta = 0.044\).

(b) The value is \(\delta = 0.031\).

Step-by-step solution

Plot the graph of the function

Draw the graph of the function\(f\left( x \right) = {\csc ^2}x\), \(f\left( x \right) = 500\) and \(f\left( x \right) = 1000\)by using the graphing calculator as:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\({\csc ^2}x\)in the\({Y_1}\)tab,\(500\)in the\({Y_2}\)tab and\(1000\)in the\({Y_3}\)tab.

2. Enter the “GRAPH” button in the graphing calculator.

The graph of the function \(f\left( x \right) = {\csc ^2}x\) is shown below:

(a)Step 2: Observe the graph

From the graph, it is observed that the graph of \(y = {\csc ^2}x\), and \(y = 500\) intersect at \(x \approx 3.186\).

Apply Precise definition of Infinite limits

Subtract \(\pi \) from \(3.186\) to determine \(\delta \).

\(\begin{aligned}\delta = 3.186 - \pi \\ \approx 0.044\end{aligned}\)

Therefore, the value of \(\delta \) is \(0.044\).

According to the Precise definition of infinite limits, if \(0 < \left| {x - \pi } \right| < 0.044\) then \({\rm{cs}}{{\rm{c}}^2}x > 500\).

(b)Step 4: Observe the graph

From the graph, it is observed that the graph of \(y = {\csc ^2}x\), and \(y = 1000\) intersect at \(x \approx 3.173\).

Apply Precise definition of Infinite limits

Subtract \(\pi \) from \(3.173\) to determine \(\delta \).

\(\begin{aligned}\delta = 3.173 - \pi \\ \approx 0.031\end{aligned}\)

Therefore, the value of \(\delta \) is \(0.031\).

According to the Precise definition of infinite limits, if \(0 < \left| {x - \pi } \right| < 0.031\) then \({\rm{cs}}{{\rm{c}}^2}x > 1000\).