Chapter 3.6, Problem 64ES

(a) Show that $\underset{x\to \pi /2}{\lim }\left(\pi /2-x\right)\tan x=1$ .

(b) Show that

$\underset{x\to \pi /2}{\lim }\left(\frac{1}{\pi /2-x}-\tan x\right)=0$

(c) If follows from part (b) that the approximation

$\tan \approx \frac{1}{\pi /2-x}$

Should be good for values of $x$ near $\pi /2$ . Use a calculator to find $\tan x$ and $1/\left(\pi /2-x\right)$ for $x=1.57$ ; compare the results.