Question

Can anyone help me with 1J-2? Not entirely sure where to start...

Answer 1

They are saying it is easy to find the derivative of cos(x) and evaluate it at x= pi/2

\[ \cos'(x)\bigg|_{x= \frac{\pi}{2} }= ?\]

The other idea is that the expression
\[ \lim_{x\rightarrow \frac{\pi}{2} } \frac{\cos(x)}{x-\frac{\pi}{2}}\]
is vaguely reminiscent of the definition of the derivative
\[ f'(x) = \lim_{\Delta\rightarrow 0} \frac{ f(x+\Delta)- f(x)}{\Delta}\]

Now if you can show that the given expression can be re-written to look like the definition of the derivative, then you can claim that it and the derivative (evaluated at pi/2) have the same value (-1 in this case)

I would begin by letting \( x = \frac{\pi}{2} + \Delta \) in the given expression.