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How does one find the Fourier Transform of 1?

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

I tried to solve it and came up with

[tex]\sqrt{\frac{2}{\pi}}\frac{1}{\omega}\lim_{t \rightarrow \infty}\sin\left(\omega t\right) [/tex]

but that is indeterminate whereas actual answer is

[tex]\sqrt{2\pi}\delta\left(\omega\right)[/tex]

So how does one actually solve this Fourier Transform.

Thanks in advance.

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

I tried to solve it and came up with

[tex]\sqrt{\frac{2}{\pi}}\frac{1}{\omega}\lim_{t \rightarrow \infty}\sin\left(\omega t\right) [/tex]

but that is indeterminate whereas actual answer is

[tex]\sqrt{2\pi}\delta\left(\omega\right)[/tex]

So how does one actually solve this Fourier Transform.

Thanks in advance.

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