Help with trig proof and limit question?? (1 Viewer)

happychappystudent45 · May 5, 2019

Hey there,

I'm having some troubles with this question here. I think I know what to do to prove the two equations but I'm not really sure if I'm doing the limit part right . Could someone give me a hand? Thanks in advance! (Question attached).

fan96 · May 6, 2019

Note that before we do anything else, $b$ must be nonzero.

Also, to apply the identity we require $1 + \cos ax \neq 0 \iff ax \neq \pm\pi, \, \pm 3\pi ...$

One method is

$\begin{aligned} & \frac{1 - \cos ax}{bx} \\ &= \frac{\sin^2ax}{bx(1+\cos ax)} \\ &= \frac{4 \sin^2\frac{ax}2\cos^2\frac{ax}2}{bx(2\cos^2\frac{ax}2)} \\ &= ... \end{aligned}$

and using the standard result

$\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$

leads to the answer.

(This method doesn't work for $a = 0$ but this case is trivial to verify separately.)

Help with trig proof and limit question?? (1 Viewer)

happychappystudent45

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