As h approaches 0,

lim [cos((pi/2)+h) - cos(pi/2)] / h

The answer is -1. Can someone please explain how to arrive at this answer?

Are there special trig-based formulas I need to know to solve questions like this?

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- May 14th 2012, 05:26 PMTWNHaving trouble with trig based limits
As h approaches 0,

lim [cos((pi/2)+h) - cos(pi/2)] / h

The answer is -1. Can someone please explain how to arrive at this answer?

Are there special trig-based formulas I need to know to solve questions like this? - May 14th 2012, 05:47 PMReckonerRe: Having trouble with trig based limits
Two special limits you should know are

$\displaystyle \lim_{x\to0}\frac{\sin x}{x}=1$ and $\displaystyle \lim_{x\to0}\frac{1-\cos x}{x}=0$.

In this case, we have

$\displaystyle \lim_{h\to0}\frac{\cos\left(\frac{\pi}2+h\right)-\cos\frac{\pi}2}h$

$\displaystyle =\lim_{h\to0}\frac{\cos\frac\pi2\cos h-\sin\frac\pi2\sin h-0}h$ (sum and difference identity)

$\displaystyle =\lim_{h\to0}\frac{0\cdot\cos h-1\cdot\sin h}h$

$\displaystyle =\lim_{h\to0}\frac{-\sin h}h$

$\displaystyle =-\lim_{h\to0}\frac{\sin h}h$

$\displaystyle =-1$ - May 14th 2012, 06:32 PMTWNRe: Having trouble with trig based limits
Ah, I see. I was unaware of the sum and difference identities. Thank you!

By the way, how do you write out the problems like that? It's so much easier to read than mine is! - May 14th 2012, 06:37 PMReckonerRe: Having trouble with trig based limits
You can write in $\displaystyle \LaTeX$, just enclose your code between [tex] and [/tex] tags. For help with LaTeX, visit the LaTeX forum. There's a tutorial there.

- May 15th 2012, 03:06 AMskeeterRe: Having trouble with trig based limits
note the format of the limit ...

$\displaystyle \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

... this is the definition of $\displaystyle f'(x)$

in this case, $\displaystyle f(x) = \cos{x}$ and the limit is $\displaystyle f'\left(\frac{\pi}{2}\right) = -\sin \left(\frac{\pi}{2}\right) = -1$ - May 15th 2012, 02:16 PMTWNRe: Having trouble with trig based limits
That makes sense! Thank you both!