lim_(n->âˆ‍) n = د€/(sin(د€/n)) is this formula common knowledge or I came up with it? If it is what is it? [on hold]

I am not mathematician, I was just observing in my thoughts what is infinity, and how could I visulize it and as a result of 2 week journey in this thoughts, me who has no experience with math concluded this:

lim_(n->âˆ‍) n = د€/(sin(د€/n))

I tested this formula and it happens to be true, so I wonder if it has something silly, untrue, or something that is know and used.

But for me it sound like an answer to the question, how many dot are there in infinity, that is me not a mathematician a regular guy.

Please help me with where I am, I was fascinated with math as I was doing this, I am 37 now and have rediscovered something that was very unclearly explained to me in school, even if was a good student, I now realize that I understood math just now. So lead me.

I can explain how I arrived to this, if needed.

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This is well-known and is easily explained by the fact that sinx≈x\sin x\approx x for small xx, so that

πsinπn≈ππn=n.\frac\pi{\sin\frac\pi n}\approx\frac\pi{\frac\pi n}=n.

Actually every smooth function can be approximated by a linear law f(x)≈a+bxf(x)\approx a+bx for small xx, with f(0)=af(0)=a. (This is the equation of the tangent at the origin.)

In the case of the sine,

sinx≈sin0+bx=bx\sin x\approx \sin 0+bx=bx and one can show that the constant bb is just 11.

Such approximations are extremely useful in many circumstances.

@rahul: nothing special, this can be shown by the Taylor development.
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