How to compute this limit involving complementary error functions I am trying to take the following limit
$$\lim_{x\to \infty } \, \frac{2 x \operatorname{erfc}\left[\frac{x}{\sqrt{2} t}\right]}{t \operatorname{erfc}\left[-\frac{x}{\sqrt{2}}\right]}$$
my first thoughts were to use LHospital's rule after making the top complementary error function as the denominator of the denominator. Upon differentiating I got the following
$$\lim_{x\to \infty } \, \frac{\sqrt{2 \pi } \exp\left[\frac{\left(t^2+1\right) x^2}{2 t^2}\right] \operatorname{erfc}\left[\frac{x}{\sqrt{2} t}\right]^2}{\exp\left[\frac{x^2}{2}\right] \left(\operatorname{erf}\left[\frac{x}{\sqrt{2}}\right]+1\right)+t \exp\left[\frac{x^2}{2 t^2} \right]\operatorname{erfc}\left[\frac{x}{\sqrt{2} t}\right]}$$
which seems even more complicated.
Maybe using an identity or some form of expansion for the complementary error function may lead to a simpler expression.
Any help will be greatly appreciated.
Thank you.

The denominator tends to $t$. Depending on the sign of $t$, the numerator tends to $\infty$ or $0$, because the complementary error function tends to $1$ or $0$, with a super-exponential rate.
