Cauchy Principle Value

This is something I thought of while reviewing the inversion formula for characteristic functions in probability theory.

Is it true that if we have a continuous and bounded function $f:mathbb{R}rightarrowmathbb{C}$ then for the following two limits, $$lim_{Trightarrowinfty}int_{-T}^Tf(x),dx$$ and $$lim_{epsilonrightarrow 0}int_{-infty}^infty f(x)e^{-epsilon^2x^2/2},$$ one exists iff the other one exists, and in case they both exists the limits agree?

This seems true for $sin$ and $cos$, is this true in general?