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 2 years ago
Really need help understanding this limit/derivative question...
The limit
Lim x approachs 5pi CosX +1 / x5pi
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...
 2 years ago
Really need help understanding this limit/derivative question... The limit Lim x approachs 5pi CosX +1 / x5pi 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...

This Question is Closed

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1Remember the limit definition of a derivative?\[\large f'(x)=\lim_{h \rightarrow 0}\frac{f(xh)f(x)}{h}\]Well there is also another definition that we see less often, of this form,\[\large f'(x)=\lim_{x \rightarrow a}\frac{f(x)f(a)}{xa}\]This is the one that we want to analyze.

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1If we compare this to the limit we were given, we can see that is looks like our A value will be 5pi yes? \[\large \lim_{x \rightarrow 5\pi}\frac{\cos x +1}{x5\pi}\]

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1The top is still a little tricky though, we have to sort it out.

MarcLeclair
 2 years ago
Best ResponseYou've already chosen the best response.0wait the second one just looks like the mean value theorem? it oddly looks like f(a)b/ab = f'(c)

MarcLeclair
 2 years ago
Best ResponseYou've already chosen the best response.0oh nevermind me theres a limit sorry im a bit tired

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1heh, yah it looks similar :)

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1\[\large \cos(5\pi)=?\]

MarcLeclair
 2 years ago
Best ResponseYou've already chosen the best response.0So 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.

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1That'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?

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1With 2 extra spins* my bad.

MarcLeclair
 2 years ago
Best ResponseYou've already chosen the best response.0I'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

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1If 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.

MarcLeclair
 2 years ago
Best ResponseYou've already chosen the best response.0i mean pi i mean cos and sin... wow I' m sounding stupid

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.15pi is the same as 180 degrees.

MarcLeclair
 2 years ago
Best ResponseYou've already chosen the best response.0so 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. :)

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1No 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,\[\large \large f'(x)=\lim_{x \rightarrow a}\frac{f(x)f(a)}{xa}\qquad\qquad \rightarrow \qquad \qquad \lim_{x \rightarrow 5\pi}\frac{\cos x +1}{x5\pi}\]

MarcLeclair
 2 years ago
Best ResponseYou've already chosen the best response.0Yeah it's represented by 5pi?

MarcLeclair
 2 years ago
Best ResponseYou've already chosen the best response.0so i have to equate both limit? and finding my h?

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1no h, we're using the second definition that i posted, not the one involving h.

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1It's another form of the limit definition. It comes up less often.

MarcLeclair
 2 years ago
Best ResponseYou've already chosen the best response.0My bad I understood we had to manipulate it back to the other form

MarcLeclair
 2 years ago
Best ResponseYou've already chosen the best response.0Then when its ask to find F, am I suppose to find cosx?

MarcLeclair
 2 years ago
Best ResponseYou've already chosen the best response.0sorry, the wording is really throwing me off

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1If 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.

MarcLeclair
 2 years ago
Best ResponseYou've already chosen the best response.0Oh wow. I understand. This was a very basic question but I have never seen that definition in my entire class. Thanks a lot! :)

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1\[\large \frac{\cos x+1}{x5\pi} \qquad =\qquad \frac{\cos x(1)}{x5\pi} \qquad = \qquad \frac{\cos x(\cos 5\pi)}{x5\pi}\]

MarcLeclair
 2 years ago
Best ResponseYou've already chosen the best response.0Thanks a lot you're patient!
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