The Limit of the Minimum of Two Random Variables...

Question

I am looking at the following random variable: $$T = \min_n{\{Q_n + X \cdot S_n\}}$$

where $Q_n$ and $S_n$ are exponentially distributed, and $X$ is a constant. I am interested in two things.

1. As $n \rightarrow \infty$, what does $T$ converge in distribution to? and,
2. As $n \rightarrow \infty$, what does $E[T]$ converge to?

I think that

1. It may converge to some type of Shifted Erlangian-2 r.v. and
2. It looks like $E[T] \rightarrow 0$ as $n \rightarrow \infty$ w.p. 1.

But, I don't know how to prove (or show) this and I may not even be correct in my conclusions above.

Any help would be appreciated.