could any1 tell me how to find the limit as i have tried every mthod i now and its getting me nowere.
Does the limit even exist?
I was thinking that as , and doesn't approach a limit i think it would just oscilate. So no limit would exist.
could any1 tell me how to find the limit as i have tried every mthod i now and its getting me nowere.
Does the limit even exist?
I was thinking that as , and doesn't approach a limit i think it would just oscilate. So no limit would exist.
This is what I would do:
Now, recall that
So we have
We should familiarize ourselves with two special limits:
and
We need to incorporate these somehow...
We can first start off by simplifying this:
Now if you make a substitution, say , we see that as ,
Thus,
Thus,
Does this make sense?
--Chris
It's overkill I guess.
We know that .
We also know that cos() is always between -1 and 1:
If an expression has limit 0, then we may multiply it with things that are maximum 1 in absolute value, the result will still have limit 0.
Why?
Say we want to get a neighborhood of 0 where f(x) is always close or even closer to 0.
We can choose such a neighborhood for sin(x), because sin(x) has limit 0 at x=0, and the definition of the limit states just what we want.
But in f(x), we also have that cos(1/x) thing there. Okay, but we saw it is between -1 and +1, so multiplying by it can only take us closer to 0, or leave us as close as before. (It might get us on the other side, if it is negative, but the distance from 0 can not be larger than before).
So? We are done. We got the neighborhood of 0, where f(x) is always maximum far from 0. We didn't use how small is, so by definition of the limit, f(x) has the limit 0.