Question

Compute the limit as X goes to zero for the function: (cosX/cos 2X)^ 1/X^2

Answer 1

\[\lim_{x \to 0}\left(\frac{\cos(x)}{\cos(2x)}\right)^{\frac{1}{x^2}}\] like that?

Answer 2

take the log, get
\[\frac{1}{x^2}\ln\left(\frac{\cos(x)}{\cos(2x)}\right)\]
then maybe simplify using properties of the log, take the limit using l'hopital, and then exponentiate

Answer 3

(cosx/cos 2x) ^ (1/x sqared). we started limits in calc but nothing with sin or cos. thats why im not sure how to even start this

Answer 4

just started? do you know l'hopital's rule?

Answer 5

not yet, he gave us this as a bonus. normally the limits we have to solve are for when x goes to + or - infinity and the problems are like [(2x^2+3x-4)/(3x^4-5x+6)]

Answer 6

i cannot do this without l'hopital. i really have no idea what he expects. the limit inside the parentheses is 1 so you have the form
\[1^{\infty}\] which is indetermined. you have to take the log, take the limit, and then exponentiate (that is, raise e to the power of whatever you get) to get your answer.
i cannot think of an elementary way to do this. but the final answer will be
\[e^{\frac{3}{2}}\] maybe i can ask someone for a simple method

Answer 7

we have gone over: f(x)=e^(-1/x) and e^-infinity =0, and e^+infinity = +infinity. so would the limit be +infinity?

Answer 8

Hm, if you don't use l'Hopital's rule, you're still going to need to use the derivative somewhere. How about this: do you know the power series expansion/Taylor series expansion/Maclaurien series expansion of cos x?

Answer 9

in fact you need l'hopital twice. hard to imagine expanding in a taylor series before l'hopital

Answer 10

i just thought maybe there was a gimmick i didn't see, but i am thinking there is not one

Answer 11

(maybe. I learnt power series in high school before we discussed l'Hopital's rule.) I don't think there is one either.

Answer 12

Good question though. But insanely hard if you don't have some of these sharp instruments in your tool box.

Answer 13

So is my professor asking us to solve a problem he knows we wont be able to solve?....yet

Answer 14

Perhaps he's doing it on purpose to help motivate the next thing he teaches you.

Answer 15

thats what i was thinking, thanks for the help. and i just learned of this web site a few weeks ago and i think its fantastic. im not sure who to tell, but it deserves the praise!

Answer 16

Glad you're finding it helpful.