## Sinc function limit question

Consider $sin(vx)/x$ as $v$ approaches infinity. Now, this becomes a delta function and I have seen graphs that show this function as v increases.

My question is the following: I cannot quite see why the function $sin(vx)/x$ becomes zero if x≠0. sin(vx) is bounded and oscillates rapidly between -1 and 1 as x is changed. But the envelope of the function is still 1/x so how come it goes to zero for all x≠0? Can someone prove this result?

Thank you.
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 I think you're talking about sin(1/x) not, (sinx)/x Edit: Actually re-reading your post, I realize I have no idea what you're asking. sinx/x is in no way related to the delta function, and does not oscillate between anything, it's just a normal sinx that diminishes as you approach infinity. Sin(1/x) however does oscillate in a divergent manner as you approach 0, the limit is quite strange and nothing is quite like it.
 No, it is sin(vx)/x. As v gets larger, it approximates a delta function better. Now, with v→∞, I cannot see why the function is zero if x is non zero. A similar definition is here http://en.wikipedia.org/wiki/Sinc_function where there is a section on how it relates to the dirac delta function.

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## Sinc function limit question

The proper way to show that

$$\lim_{v \rightarrow \infty} \frac{\sin(vx)}{x} = \delta(x)$$

is not to try and show that the left hand side is infinite when x = 0 and zero for non-zero x. That might be true of some representations of the delta function, but really what you need to show is that

$$\lim_{v\rightarrow \infty} \int_{-\infty}^\infty dx~\frac{\sin(vx)}{x} f(x) = f(0).$$

Similarly you can show that if the integration region does not contain the origin, the integral is zero.

That is the appropriate sense in which one should interpret "$\lim_{v \rightarrow \infty} \frac{\sin(vx)}{x} = \delta(x)$".
 Thank you Mute. Didn't know that before but yeah, I can prove it according to your definition so I guess it's all good.

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