Thread: Limit of a sequence

Hi everyone,

Prove the following:


I must show that for each we can find such that for every .

Thing is that I can't figure out how to "get rid of" . Is bigger than anything for every ?
Thanks!

For small x, sinx is approximately equal to x. So as n increases sin(1/n) is approximately equal to 1/n

True, but this doesn't really help me. I need to get rid of by showing that it's bigger than something else, not smaller, no?

I was thinking of replacing sin(1/n) by 1/n and then the limit is clearly infinite, but I suspect that may not do as a solution.

Yes you can approximate by as n tends to infinity.There is nothing wrong in that.
As an example to enlighten you,even can be approximated by as n tends to infinity.

Yes but how does that help me? I need to prove the limit by definition.
For example, I can do this:



but then what?

Originally Posted by loui1410

Yes but how does that help me? I need to prove the limit by definition.
For example, I can do this:



but then what?

Now given any M can you find your N?Think about it.

but how did you get to

?

This is the move I'm looking for, but why is this true?

There are many ways to prove it.But simplest is to look at the Taylor series exapansion of sin(x).

We haven't learned Taylor series yet so I'm not supposed to use it, there must be another solution that they had in mind. Anyway, I'll ask the teaching assistant tomorrow for guidance and see what happens..