Fourier Transform: Limit in Infinity of Exponential Function

[aaaa202]

In calculating some basic Fourier transform I seem stumble on the proble that I don't know how to take the limit in infinity of an exponentialfunction with imaginary exponent. In the attached example it just seems to give zero but I don't know what asserts this property. I would have thought that it would yield something infinite since a cosine or sine does not go to zero at infinity. What is done to arrive at the attached result?

 

[fluidistic]

In calculating some basic Fourier transform I seem stumble on the proble that I don't know how to take the limit in infinity of an exponentialfunction with imaginary exponent. In the attached example it just seems to give zero but I don't know what asserts this property. I would have thought that it would yield something infinite since a cosine or sine does not go to zero at infinity. What is done to arrive at the attached result?

Think of $e^{(ik-a)x}$ as equal to $e^{ikx} \cdot e^{-ax}$. On the right hand side while $e^{ikx}$ is not really defined for when x tends to infinity, it is still bounded (because sine and cosine are bounded) and the term $e^{-ax}$ does tend to 0. So that the product tends to 0 and you get $\lim _{x \to \infty}e^{(ik-a)x} =0$.