Black-Scholes PDE and its limits in spot

Starting from geometric Brownian motion for a stock, derive the Black-Scholes partial differential equation for the price of a derivative on it. What conditions does a call's price satisfy as the stock price tends to $0$ and to $\infty$?

$V_t+\tfrac12\sigma^2S^2V_{SS}+rSV_S-rV=0$. Apply Itô's lemma to $V(S,t)$, hold the option short against $V_S$ shares so the $dW$ terms cancel, and require the riskless portfolio to earn $r$. For a call, $C\to0$ as $S\to0$, and $C\sim S-Ke^{-r(T-t)}$ as $S\to\infty$, with $C(S,T)=(S-K)^+$ at expiry.