Weight aspect asymptotic and simultaneous non-vanishing for Rankin-Selberg $L$-functions

In this article we show simultaneous non-vanishing of two Rankin-Selberg $L$-functions by proving an asymptotic result in weight aspect. The main input of this paper is to remove the $t$-integral dependence from the result of Blomer-Harcos (see \cite{BH2}) and getting a square root exponent for the error term.