Analyzing the limit $\lim_{t\rightarrow\infty}\int_{0}^{t}f_1(t,\tau)f_2(\tau)\mathrm{d}\tau.$

Question

I am studying the function $$f(t)=\int_{0}^{t}f_1(t,\tau)f_2(\tau)\mathrm{d}\tau.$$ I known that the limit $\lim_{t\rightarrow\infty}f_1(t,\tau)=g(\tau)$ exists and is finite. Is it possible to prove that $\lim_{t\rightarrow\infty}f(t)=\int_{0}^{\infty}g(\tau)f_2(\tau)\mathrm{d}\tau$ under the assumption that this limit exists?