If I have two exponential function, say $f_1(t)=4e^{-3t}+6e^{-7t}$ and $f_2(t)=\frac{2e^{-3t}+5e^{-7t}}{e^{-3t}+9e^{-7t}} - 2$ who are all converge to $0$. Then, the convergence rate of $f_1(t)$ can be quantified by $min\{|-3|,|-7|\}=3$, and for $f_2(t)$ we can use $|(-7)-(-3)|=4$.
Is it reasonable to say $f_2(t)$ converges faster than $f_1(t)$? Thanks!
Notice that
$$f_1(t)\sim_\infty4e^{-3t}$$ and $$f_2(t)=-\frac{13e^{-7t}}{e^{-3t}+9e^{-7t}}\sim_\infty-13e^{-4t}$$ so $$f_2(t)=_\infty o(f_1(t))$$
hence $f_2(t)$ converges faster than $f_1(t)$ to $0$.