I am considering the dynamics $$ dS_t = (r-q)S_t dt + \sigma(t,S_t)S_t dW_t. $$ For a Call Option $C$ with strike $K$ I am supposed to derive the equation $$ \frac{K^2}2\sigma^2(T,K)\partial_{KK}C = \partial_TC + (r-q) K\partial_KC + qC. $$ Using the Fokker-Planck equation $$ \partial_tf(x,t) + (r-q)\partial_xxf(x,t) = \frac12\partial_{xx}\left[\sigma^2(t,x) x^2f(x,t)\right]$$ for the density of the underlying, I was able to derive the equation using partial integration (I did it pretty much like here, section 2). At one point, while using partial integration, one has to deal with the limits (see section 2.4.1 and 2.4.2 in the above link) $$ \lim_{x \to \infty} x(x-K)f(x,T) \quad \text{and} \quad \lim_{x \to \infty}\partial_{x}\left[\sigma^2(t,x) x^2f(x,t)\right] $$ Everything works out if these limits are 0 and this is also assumed in the link. However, I tried to argue rigorously that these limits should be 0, but I have no clue where to even start. I am not even sure if that is even true in general.
Can you provide me with some hint on how to prove it (possibly using the Fokker Planck equation) or give me some idea on how to create a counterexample (i.e. some $r,q$ and $\sigma(t,S_t)$ for which the above equation does not hold.)
Thanks in advance! (I was not sure whether I should post this on Quant Stack Exchange, because the topic is more about the stochastic/partial differential equations and not so much about the underlying finance.)