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

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Homework Help Overview

The discussion revolves around evaluating the limit of the expression tan(ax)/x as x approaches infinity, within the context of quantum physics. Participants are exploring the behavior of trigonometric functions and their limits.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the relationship between the sine and cosine functions and their behavior as x approaches infinity. Questions arise regarding the convergence of these functions and the implications for the limit of the overall expression. There is also a mention of the need for a theorem related to the product of functions approaching zero.

Discussion Status

The discussion is active, with participants providing insights and questioning assumptions about the convergence of trigonometric functions. Some guidance has been offered regarding the behavior of tan(ax) and its implications for the limit, though there is no explicit consensus on the approach to the problem.

Contextual Notes

One participant notes a misunderstanding regarding the function involved, clarifying that it was tanh instead of tan, which changes the nature of the limit being discussed. This highlights the importance of correctly identifying functions in limit problems.

maria clara
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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!
 
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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.
 
it does not converge
 
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!
 
maria clara said:
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! :smile:

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?
 
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...:)
 

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