Thread: General Sequence Limits Question and Method Know How

If searched the forums for the explanation of the famous

• $\displaystyle \lim_{n\to\infty}\frac{sin(n)}{n}$

I am not saying the explainations in some thread weren't concrete, its just something about this limit or I am just over complicating a simple things here, I guess teachers like to use this example because the $\displaystyle sin(n)$ seems to throw people off and this limit infact behaves similarily to just $\displaystyle \frac{1}{n}$ It seems like I am just complicating things that are obvious, please feel free not to waste your time and link me to an explanation if you'd like.

Also my other question, how do you know what theorem or tests works the best for the limits or convergence of a sequence function, as in what are the mains things you look for?

Originally Posted by RockHard

If searched the forums for the explanation of the famous

• $\displaystyle \lim_{n\to\infty}\frac{sin(n)}{n}$

I am not saying the explainations in some thread weren't concrete, its just something about this limit or I am just over complicating a simple things here, I guess teachers like to use this example because the $\displaystyle sin(n)$ seems to throw people off and this limit infact behaves similarily to just $\displaystyle \frac{1}{x}$ It seems like I am just complicating things that are obvious, please feel free not to waste your time and link me to an explanation if you'd like.

Also my other question, how do you know what theorem or tests works the best for the limits or convergence of a sequence function, as in what are the mains things you look for?

Squeeze theorem. Look it up. We may bound this function by $\displaystyle \left|\frac{\sin(n)}{n}\right|\le\frac{1}{n}$ and since the RHS converges to zer so must the LHS.

Can the squeeze theorem be used in "proving" the limit of that function?
Once again I over complicate things, but there seems to be so many theorems and rules and etc. its over whelming when to use what or even recall a concept of a theorem, practice makes perfect a las. Thanks again drexel

Originally Posted by RockHard

Can the squeeze theorem be used in "proving" the limit of that function?
Once again I over complicate things, but there seems to be so many theorems and rules and etc. its over whelming when to use what or even recall a concept of a theorem, practice makes perfect a las. Thanks again drexel

Yes. You can rigorously prove the squeeze theorem and apply it with reckless abandon. Or just note that if you choose $\displaystyle n>\frac{1}{\varepsilon}$ that

$\displaystyle \left|\frac{\sin(n)}{n}-0\right|\le\frac{1}{n}<\frac{1}{\tfrac{1}{\varepsi lon}}<\varepsilon$

Ahhh, ok it is starting to make some sense, if I look at it this way.
The function $\displaystyle sin({x})$ is bounded by $\displaystyle -1\le{sin(x)}\le{1}$ then the statement $\displaystyle \frac{-1}{x}\le\frac{{sin(x)}}{x}\le\frac{{1}}{x}$ is true and converges to 0 for any $\displaystyle x \ge 0$?
Similar with the function function $\displaystyle cos(x)$ is bounded the same and the squeeze theorem can hold true for that as well? Can it hold for any function over just some x value as x approaches positive infinity?