Solve Quantum Physics Limit: tan(ax)/x → 0 as x→∞

[maria clara]

Hi, I've been trying to solve a problem in quantum physics, and got stuck because of a limit. I guess I'm a little rusty on that and would appreciate any help.
How can I show that the expression tan(ax)/x tends to zero in the limit x---> infinity?
thanks!

 

[rock.freak667]

tan(ax)=sin(ax)/cos(ax)

thus tan(ax)/x=sin(ax)/xcos(ax)

as x-> infinity, does sin(ax) and cos(ax) approach any single value?
If it does, then you can find the answer easily

If it doesn't, then your limit would just depend on the 'x' in the denominator if you understand what I am saying.

 

[boombaby]

it does not converge

 

[maria clara]

well, that's exactly the problem, sin and cos don't approach a certain value at infinity, and 1/x does. But is there a theorem that states that if a function approches zero and another function does not approach any specific value, then the product of both would approach zero? I don't think so, there has to be a better explanation. If I'm wrong, please correct me, thanks again!

 

[tiny-tim]

Hi, I've been trying to solve a problem in quantum physics, and got stuck because of a limit. I guess I'm a little rusty on that and would appreciate any help.
How can I show that the expression tan(ax)/x tends to zero in the limit x---> infinity?
thanks!

Hi maria clara!

I think boombaby is right …

tan keeps hitting ±∞ every π/a …

can you find an N such that, for x > N, |tan(ax)/x| is always < 1, for examle?

 

[maria clara]

you're right, it doesn't converge.

and I just found out that it was all my mistake, it was tanh and not tan... and since tanh is bounded at infinity, tanh(ax)/x definitely approaches zero...

sorry guys.. thanks for your help anyway...:)