At first, we change the form of limit by replacing $x=X+1$ and $y=Y+2$ that result that
$$
\lim \limits_{(x,y) \to (1,2)}\frac{xy-2x-y+2}{x^2 + y^2 - 2x -4y + 5}=
\lim \limits_{(X,Y) \to (0,0)}\frac{X\,Y}{X^2+Y^2}
$$
Now, One of the clasical method for obtaining the limit of function with two variables in $(0,0)$ is by this fact that you can
assume $X=r \, \cos(\theta)$ and $Y=r \, \sin(\theta)$, then calculate the limit with new values for $X$ and $Y$ when $r \to 0$, if after calculation limit, the final values of limit depend on to $\theta$, we conclude that, the function dose not have limit in $(0,0)$ but if the final values of limit be independent of $\theta$, we say the function has limit in $(0,0)$ with value that we obtained. so we have
$$\lim_{(X,Y) \to (0,0)}\,\dfrac { X\,Y } {X^{2}+Y^{2} }
=\lim_{(r) \to (0)}\frac{r^2\,\cos(\theta)\, \sin(\theta)}{r^2}=\cos(\theta)\, \sin(\theta)
$$
that means your limit does not exist in $(0,0)$.