Question

Basic limit question

Answer 1

I need to proof this:

\[\lim_{x \rightarrow 0}(x^{4}\cos \frac{ 2 }{ x })\] is zero

Answer 2

I separated the limits, the first one is zero and zero * a limit is zero...can I do that?

Answer 3

you can separate the limits only when both the limits exist

Answer 4

Oh...then I'm in problems...let me try for a min

Answer 5

lim of cos(2/x) is indeterminate but we know its bound between -1 and 1

Answer 6

are you allowed to use taylor series

Answer 7

Ok...so what should I do? (I can take out the 2 because it is a constant...but I'm not sure how to follow)

Answer 8

Nope...I mean, I could use this numerically or with a graph. I just wanted to try algebraic way

Answer 9

use squeeze them
recall
-1=<cos(2/x)<=1

Answer 10

So I just say it's 0?

Answer 11

well show the inequality you work with and show how the squeeze thm applies

Answer 12

like you can multiply all sides of that inequality by a little function i liked to call x^4 i gave so it works for this situation

Answer 13

And x^4 is positive so the inequality does not change if you multiply thoroughout by x^4.

Answer 14

nice freckles you still remember all nice things in calc :) this theorem is literally like squeezing :
\[\large -1\le \cos(2/x) \le 1 \\ ~\\\large \implies \lim\limits_{x\to 0}-x^4 \le~ \lim\limits_{x\to 0}x^4\cos(2/x) ~\le \lim\limits_{x\to 0}x^4 \]

Answer 15

So it's 0 :) thanks

Answer 16

it is 0
that is what you were trying to prove right?
so the proof is actually the answer

Answer 17

Ok :) thanks to everyone. I really did not understand the squeeze theorem...until now

Answer 18

Cats for you! http://tinyurl.com/lcebbyk

Answer 19

http://assets.openstudy.com/updates/attachments/5404d2ece4b0f2ed1e147305-bswan-1409629617917-ff.gif

Answer 20

One question. If the limit was 0.5, I would've not gotten the same answer. Would it still be correct?

Answer 21

Oh...I just saw the graph :)