Question

Using the fact that lim h->0 sin(h)/h = 1
and lim h->0 cos(h)-1/h =0
compute the following limits
a. lim h->0 sin(x+h)-sin(x)/h

b lim h->0 cos(x+h)-cos(x)/h

Answer 1

addition angle formula for the first one
second one too

Answer 2

\[\sin(x+h)-\sin(x)=\sin(x)\cos(h)+\sin(h)\cos(x)-\sin(x)\]
divide all by \(h\) and take the limit

Answer 3

you get that?

Answer 4

I think how about b

Answer 5

exactly the same, but this time the addition angle formula for cosine

Answer 6

i can walk you through one if you like
the other is the same

Answer 7

satelline, you will do all of that work?

Answer 8

btw do you know what the answer is before you begin?
@SolomonZelman what work?

Answer 9

k please

Answer 10

to which one ? Also, I am logging off in 2 minutes... SCHEDULE :P

Answer 11

sorry

Answer 12

lets do the first one

Answer 13

k

Answer 14

I am not a good teacher anyway ... bye

Answer 15

better than me

Answer 16

before we begin, is it clear that you are being asked for the derivative of sine, and that the answer is cosine? to the first one i mean

Answer 17

yes

Answer 18

k
lets ignore the limit for the moment so i don't have to write it in, and just to the algebra first

Answer 19

\[\frac{\sin(x+h)-\sin(x)}{h}=\frac{\sin(x)\cos(h)+\sin(h)\cos(x)-\sin(x)}{h}\] is a start
that by the "addition angle formula for sine

Answer 20

now break in to two parts
\[\frac{\sin(x)\cos(h)-\sin(x)}{h}+\frac{\sin(h)\cos(x)}{h}\] factor the first one and get
\[\sin(x)\frac{\cos(h)-1)}{h}+\frac{\sin(h)\cos(x)}{h}\]

Answer 21

now take the limit as \(h\to 0\)

Answer 22

since \[\lim_{h\to 0}\frac{\cos(h)-1}{h}=0\] the first term goes bye bye and since
\[\lim_{h\to 0}\frac{\sin(h)}{h}=1\] the second term becomes \(\cos(x)\) and needed

Answer 23

I think I got it but what formula for b?

Answer 24

the addition angle for cosine is
\[\cos(x+h)=\cos(x)\cos(h)-\sin(x)\sin(h)\]

Answer 25

do you have time to show the work?

Answer 26

i have the time, but why don't you try it first
if you mimic what i did above, you will
a) understand it better and
b) learn what it is about it you don't understand

then ask

Answer 27

k thanks so much

Answer 28

yw
write if you get stuck