Under which conditions
$\lim_{a\to+\infty}\ln(f(a,x)) = \ln(z(x))\Longrightarrow \underset{a\to+\infty}{\lim}f(a,x) = z(x)\;?$
Under which conditions
$\lim_{a\to+\infty}\ln(f(a,x)) = \ln(z(x))\Longrightarrow \underset{a\to+\infty}{\lim}f(a,x) = z(x)\;?$
It's always true. You can take exponentials on both sides, and then pull the $\lim$ outside on the left-hand side since the exponential function is continuous.