Win-by-N match as the margin grows

Player A wins each round against Player B independently with probability $0.6$. They play until one of them is ahead by $N$ rounds, and that player takes the match. What does the probability that A takes the match tend to as $N\to\infty$?

$1$. By the gambler's ruin formula with odds ratio $r=0.4/0.6=\tfrac23$, A reaches a lead of $N$ before B does with probability $\tfrac{1}{1+r^N}$, which tends to $1$ as $N$ grows.