Question

Really need help understanding this limit/derivative question...
The limit
Lim x approachs 5pi CosX +1 / x-5pi


represents the derivative of some function f(x) at some number a. Find f and a .

I don't even understand the wording of this question...

Answer 1

Remember the limit definition of a derivative?\[\large f'(x)=\lim_{h \rightarrow 0}\frac{f(x-h)-f(x)}{h}\]Well there is also another definition that we see less often, of this form,\[\large f'(x)=\lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a}\]This is the one that we want to analyze.

Answer 2

If we compare this to the limit we were given, we can see that is looks like our A value will be 5pi yes?
\[\large \lim_{x \rightarrow 5\pi}\frac{\cos x +1}{x-5\pi}\]

Answer 3

The top is still a little tricky though, we have to sort it out.

Answer 4

wait the second one just looks like the mean value theorem? it oddly looks like f(a)-b/a-b = f'(c)

Answer 5

oh nevermind me theres a limit sorry im a bit tired

Answer 6

heh, yah it looks similar :)

Answer 7

\[\large \cos(5\pi)=?\]

Answer 8

So yeah, but we're looking for the original function is that it or looking for the derivative? and cos5pi would be... .96 ? But I think i did it in degree so it might be wrong.

Answer 9

That's one of your special angles that you're going to want to remember.
It will produce the same value as Pi. 5pi is Pi with an extra spin around the circle.
-1 yes?

Answer 10

With 2 extra spins* my bad.

Answer 11

I'll ask you one thing before we continue, when taking pi in a derivative, do we ALWAYS use it in radians? I mean sometimes it works when I don't put it in my calculator as a radian

Answer 12

If you're dealing with Pi, then yes you need to be in radians :o
You could convert to degrees if radians are confusing you though.

Answer 13

i mean pi i mean cos and sin... wow I' m sounding stupid

Answer 14

5pi is the same as 180 degrees.

Answer 15

so in the question am I using L'hospital rule or finding the limit? that's where I'm lost. the question is really confusing . And alright thanks for explaining the pi. :)

Answer 16

No L'Hop.
We're relating a weird looking limit back to the Limit Definition of a Derivative.
We need to match up the pieces so we can see what the original function was.

So far we've established that our A value is 5pi. If you're unsure about that, compare the form of our limit with the Definition,\[\large \large f'(x)=\lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a}\qquad\qquad \rightarrow \qquad \qquad \lim_{x \rightarrow 5\pi}\frac{\cos x +1}{x-5\pi}\]

Answer 17

See the a?

Answer 18

Yeah it's represented by 5pi?

Answer 19

so i have to equate both limit? and finding my h?

Answer 20

no h, we're using the second definition that i posted, not the one involving h.

Answer 21

It's another form of the limit definition. It comes up less often.

Answer 22

My bad I understood we had to manipulate it back to the other form

Answer 23

Then when its ask to find F, am I suppose to find cosx?

Answer 24

sorry, the wording is really throwing me off

Answer 25

If we can show that the limit matches the DEFINITION, then we can show what our F is.

So we've established that 5pi matches the A we're looking for.
We've also shown that cos(5pi)=-1.

If we can somehow find a -1 in the top of that fraction, we can make it look like the Definition.

Answer 26

Oh wow. I understand. This was a very basic question but I have never seen that definition in my entire class. Thanks a lot! :)

Answer 27

\[\large \frac{\cos x+1}{x-5\pi} \qquad =\qquad \frac{\cos x-(-1)}{x-5\pi} \qquad = \qquad \frac{\cos x-(\cos 5\pi)}{x-5\pi}\]

Answer 28

Make sense? :) k cool!

Answer 29

Thanks a lot you're patient!