Comparing the limit of symmetrical sequences of oscillatory integrals

Question

Cosidering the limits as $\,n \, \rightarrow \, \infty$ of the following two sequences of integrals:
$$ A_n \, = \,\int_{-\infty} ^{\infty} f(ix)\frac{n^{a+ix}}{a-ix}dx \; \; \; \; \; \; \; B_n \, = \,\int_{-\infty} ^{\infty} f(ix)\frac{n^{a+ix}}{a+ix}dx $$ Let us suppose that about the complex valued function of complex variable $f(ix)$ we only know that it is such that both said limits exists, with the first one vanishing: $$ \lim_{n\to\infty} \; A_n \; = \;0 $$ would it then be a completely wrong intuition to expect that $$ \lim_{n\to\infty} \; B_n \; \neq \;0 \;\;\;?$$ I try to explain said intuition starting from the observation that $$ a+ix = (a-ix)\;e^{\, i \,2 \arctan{\frac{x}{a}}} $$ Thus, while trying to visualize both integrals through their respective Riemann sums approximation we would observe a "segmented path" of infinitesimal vectors $$ f(ix)\frac{n^{a+ix}}{a-ix} \, \Delta x $$ being laid down one after the other, except that for the second integral the corresponding "segmented path" would be composed of the very same individual infinitesimal vectors, albeit each one rotated by an angle $ - 2\arctan{\frac{x}{a}}$. The intuition is that said two "segmented paths" could not possibly converge, in the limit $\,n \, \rightarrow \, \infty$, to the same point. But because I do not feel at all confident about said intuition, are similar types of results already known from the theory of oscillatory integrals? perhaps with additional hypotheses on $f(ix)$ ? Thanks to whoever may help.