For each $x>0$ let $M(x)$ be a real-valued random variable and set $M(0)=0$. Assume the random function $M(x)$ is monotone non-decreasing on $[0,\infty)$. Define $T(y)=\inf\{x\geq 0:M(x)\geq y\}$. Suppose that $e^{-y}T(y)$ converges in law to an Exponential$(\lambda)$ random variable when $y\to\infty$. Find non-random $a(x)$ and $b(x)$>0 such that $(M(x)-a(x))/b(x)$ converges in law for $x\to\infty$ to a non-degenerate random variable. Provide the distribution function of the limit random variable and name the distribution.
My attempt: Let $\xi\sim $Exponential$(\lambda)$, then $\ln(T(y))-y\to \ln\xi$ as $y\to \infty$. Substitute $y$ by $M(x)$ we have $\ln(T(M(x))-M(x)\approx \ln(x)-M(x)\to \ln \xi$ as $x\to \infty$. I am not sure if I am correct because the ending result looks like Gumbel distribution but not exactly is. Any help/clarification would be appreciated.