Exponential limit of surviving geometric branching ($P(X_n/n>x\mid X_n>0)\to e^{-x}$)

For the stated critical geometric offspring law, the conditional upper tail is exactly $[n/(n+1){]}^{\lfloor nx\rfloor }$ for $x\ge 0$. Its logarithm tends to $-x$, proving an exponential distribution limit of rate one without a transform continuity theorem. The unconditional mass at zero tends to one, so survival conditioning changes the relevant limiting population scale.