Question

Find the rates of convergence of the following functions as h → 0.
lim
h→0
sin (h − h cos h) /h
= 0

Answer 1

$$\lim_{h\to0} \frac{sin(h-hcos(h))}{h}$$

So, we know that

$$\lim_{h\to0} \frac{sin(h)}{h}=1$$

Let's multiply the numerator and denominator by (h-hcos(h)):

$$\lim_{h\to0} \frac{sin(h-hcos(h))*(h-hcos(h))}{h*(h-hcos(h))} = (1)$$ $$\lim_{h\to0}\frac{sin(h-hcos(h))}{h-hcos(h)}=1$$

$$(1) = \lim_{h\to0} \frac{h-hcos(h)}{h}=\lim_{h\to0}1-cos(h)$$

Use the Maclaurin series for cosine:

$$cos(h) = 1-\frac{h^{2}}{2!}+\frac{h^{4}}{4!}-\frac{h^{6}}{6!}+...$$

$$1-cos(h)=1-(1-\frac{h^{2}}{2!}+...)=\frac{h^{2}}{2}+...=O(h^{2})$$

Answer: the function has quadratic convergence.