Question

(a) Use a graph of

\(f\left( x \right) = {\left( {{\bf{1}} - \frac{{\bf{2}}}{x}} \right)^x}\)

to estimate the value of \(\mathop {{\bf{lim}}}\limits_{x \to \infty } \,\,f\left( x \right)\) correct to two decimal places.

(b) Use a table of values of \(f\left( x \right)\) to estimate the limit to four decimal places.

Short answer

(a) \(\mathop {\lim }\limits_{x \to \infty } f\left( x \right) = 0.14\)

(b) \(\mathop {\lim }\limits_{x \to \infty } f\left( x \right) = 0.135335\)

Step-by-step solution

Step 1:Estimate the value of\(\mathop {{\bf{lim}}}\limits_{x \to \infty } \,\,f\left( x \right)\)by using a graph

Use the following steps to plot the graph of given functions in the Symbolab graphing calculator.

1. In the graphing calculator, enter the function \({\left( {1 - 2/x} \right)^x}\).
2. The graph will show the curve and its asymptote.

It can be observed from the figure that as x is approaching, infinity \(f\left( x \right)\) has some finite value which is equal to 0.14.

So, the value of the expression \(\mathop {\lim }\limits_{x \to \infty } f\left( x \right)\) is 0.14, as \(y = 0.14\) is an asymptote to the curve of \(f\left( x \right)\).

Estimate the value of the \(\mathop {{\bf{lim}}}\limits_{x \to \infty } \,\,f\left( x \right)\) using table

The table below represents the values of \(f\left( x \right)\).

\(x\)	\(f\left( x \right) = {\left( {1 - \frac{2}{x}} \right)^x}\)
10,000	0.135308
100,000	0.135333
1,000,000	0.135335

From the table, the value of the expression\(\mathop {\lim }\limits_{x \to \infty } f\left( x \right)\) is approximately 0.135335.