Time average of product of two functions

Question

Suppose the functions $e(x)$ and $h(x)$ with Fourier transforms $E(k)$ and $H(k)$. What is the time average A of $e(x)h(x)$ in function of their Fourier transforms $E(k)$ and $H(k)$?

My attempt: \begin{align} A &= \lim\limits_{X\to \infty} \frac{1}{X} \int\limits_{-X}^{X} e(x)h(x) dx \\ &= \lim\limits_{X\to \infty} \frac{1}{X} \int\limits_{-X}^{X} \int\limits_{-\infty}^{\infty} E(k)e^{ikx} dk \int\limits_{-\infty}^{\infty} H(k')e^{ik'x} dk' dx \\ &= \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty}E(k) H(k')\left[ \lim\limits_{X\to \infty} \frac{1}{X} \int\limits_{-X}^{X}e^{i(k+k')x} dx \right] dk dk' \,, \end{align}

using Wolfram Alpha $$\lim\limits_{X\to \infty} \frac{1}{X} \int\limits_{-X}^{X}e^{i(k+k')x} dx = 0\,,$$

and thus the average $A=0$ for all functions $e(x)$ and $h(x)$... Obviously, this cannot be the case, so I have made a mistake somewhere but I cannot find it...

Answer 1

$$ \int_{-X}^{X} \; \mathrm{e}^{\mathrm{i} (k + (-k))x} \,\mathrm{d}x = 2X \text{.} $$