Question

What is lim (cos((pi/2)+h) - cos(pi/2))/h?
h->0

Answer 1

Hint :

Let \(f(x)=\cos(x)\)
the given expression is equivalent to \(f'(\pi/2)\)

Answer 2

\[\frac{ \sin(h)-0}{ h }\]

Answer 3

\[\frac{ \sin(h) }{ h }\]

Answer 4

\[\large \frac{ \color{red}{-}\sin(h)-0}{ h }\]

right ?

Answer 5

because \(\cos(\pi/2 + x) = - \sin(x)\)

Answer 6

ohhh ok! Is that like a identity?

Answer 7

do you remember angle sum identity ?
\(\cos(A+B)\)

Answer 8

sort of, it sounds familiar

Answer 9

\(\cos(A+B) = \cos A\cos B - \sin A\sin B\)

Answer 10

use that to work out \(\cos(\pi/2+h)\)

Answer 11

ohhhhhhh that makes so much sense now

Answer 12

\[\frac{ -\sin(h) }{ h }\]

Answer 13

So do I substitute 0 for h now?

Answer 14

substitute h=0 and see what you get

Answer 15

Oh yea that doesn't work with 0 on the bottom oops!

Answer 16

right, try using the standard limits

Answer 17

So -sin(h)/h
=-1?

Answer 18

Yes :

\[\large \lim\limits_{h\to 0}~\dfrac{ -\sin(h) }{ h } =- \lim\limits_{h\to 0}~\dfrac{ \sin(h) }{ h } =-1 \]

Answer 19

Ohh I understand it all now!!! Thank you so much for teaching me you're great! :)

Answer 20

np :)