The default Finance.Solver: a safeguarded Newton-Raphson (the classic
rtsafe).
It brackets the root first, then each iteration takes a Newton step when that step lands inside the bracket and is shrinking the interval fast enough, and a bisection step otherwise. This keeps Newton's quadratic speed on well-behaved flows while retaining bisection's guaranteed convergence — in one pass, rather than running Newton to exhaustion and then bisecting separately.
Because the maintained bracket always encloses a sign change, the result is a genuine root rather than a stalled non-root, and a long-dated flow whose raw Newton step would overflow simply takes a bisection step instead.