Question

15-18 Prove the statement using the \(\varepsilon \), \(\delta \)definition of a limit and illustrate with a diagram like figure 9.

15. \(\mathop {{\bf{lim}}}\limits_{x \to {\bf{4}}} \left( {\frac{{\bf{1}}}{{\bf{2}}}x - {\bf{1}}} \right) = {\bf{1}}\)

Short answer

The given limit is true.

Step-by-step solution

Step 1:Assume the values of \(\varepsilon \) and \(\delta \)

Let \(\varepsilon > 0\) and \(\delta > 0\) such that if \(0 < \left| {x - 4} \right| < \delta \), then it can be represented as;

\(\begin{aligned}\left| {\left( {\frac{1}{2}x - 1} \right) - 1} \right| < \varepsilon \\\left| {\frac{1}{2}x - 2} \right| < \varepsilon \end{aligned}\)

Solve the inequality in step 1

The inequality\(\left| {\frac{1}{2}x - 2} \right| < \varepsilon \)can be solved as:

\(\begin{aligned}{r}\left| {\frac{1}{2}x - 2} \right| < \varepsilon \\\frac{1}{2}\left| {x - 4} \right| < \varepsilon \\\left| {x - 4} \right| < 2\varepsilon \end{aligned}\)

Consider \(\delta = 2\varepsilon \), then from the inequality \(0 < \left| {x - 4} \right| < \delta \):

\(\left| {\left( {\frac{1}{2}x - 1} \right) - 1} \right| < \varepsilon \)

Thus, the value of the expressionis \(\mathop {\lim }\limits_{x \to 4} \left( {\frac{1}{2}x - 1} \right) = 1\).

Illustrate the limit using the diagram

Use the following steps to plot the graph of given functions:

1. In the graphing calculator, select “STAT PLOT” and enter the equation \(\frac{1}{2}X - 1\) in the \({Y_1}\) tab.
2. Enter the graph button in the graphing calculator.

The figure below illustrates the limit using the diagram.

Thus, the diagram is obtained.