Answer
(a) We could estimate that $\lim\limits_{x \to \infty}f(x) = 0.13$
(b) We could estimate that $\lim\limits_{x \to \infty}f(x) = 0.1353$
Work Step by Step
(a) $f(x) = (1-\frac{2}{x})^x$
By using the zoom function on a graphing calculator, we can see that the value of the function is approximately $0.13$ as the values of $x$ become more positive.
We could estimate that $\lim\limits_{x \to \infty}f(x) = 0.13$
(b) $f(x) = (1-\frac{2}{x})^x$
We can evaluate the function at various values of $x$:
$f(10) = (1-\frac{2}{10})^{10} = 0.1074$
$f(100) = (1-\frac{2}{100})^{100} = 0.1326$
$f(500) = (1-\frac{2}{500})^{500} = 0.1348$
$f(1000) = (1-\frac{2}{1000})^{1000} = 0.1351$
$f(5000) = (1-\frac{2}{5000})^{5000} = 0.1353$
$f(10,000) = (1-\frac{2}{10,000})^{10,000} = 0.1353$
We could estimate that $\lim\limits_{x \to \infty}f(x) = 0.1353$
