Calculate the Fourier Transform of the function

Question

I have to calculate the Fourier transform of the function $f(x)=sgn(x) e^{-a |x|}, a \geq 0$. After that I have to take the limit $ a \rightarrow 0$ to calculate the Fourier transform of the sign function $f(x)=sgn(x)$. $$$$ I have found that the Fourier transform of the function $f(x)=sgn(x) e^{-a |x|}$ is: $$\widetilde{f}(k)=- \frac{2ik}{a^2+k^2}$$ Is this correct??? $$$$ Then to calculate the Fourier transform of the sign function, can I take the limit $ a \rightarrow 0$ at $\widetilde{f}(k)=- \frac{2ik}{a^2+k^2}$?? Or do I have to calculate the integral from the beginning??