How can duality be used to solve for the Fourier Transform of a constant?

[dimension10]

How does one find the Fourier Transform of 1?

$$\mathscr{F}\{1\}=\mathcal{F}\{1\}=\int\limits_{-\infty}^{\infty}{e}^{-i \omega t} \mbox{d}t=?$$

I tried to solve it and came up with

$$\sqrt{\frac{2}{\pi}}\frac{1}{\omega}\lim_{t \rightarrow \infty}\sin\left(\omega t\right)$$

but that is indeterminate whereas actual answer is

$$\sqrt{2\pi}\delta\left(\omega\right)$$

So how does one actually solve this Fourier Transform.

Thanks in advance.

 

[homeomorphic]

Use duality. Compute the Fourier transform of the delta function and you get a constant, so the Fourier transform of a constant is a delta function. Just look up duality for Fourier transforms and you'll see what I mean.

 

[dimension10]

Use duality. Compute the Fourier transform of the delta function and you get a constant, so the Fourier transform of a constant is a delta function. Just look up duality for Fourier transforms and you'll see what I mean.

Thanks a lot!