Find the value of $c$ to make $f_{XY}(x,y)$ a valid joint pdf $f_{XY}(x,y) = cx $ where $x>0$, $y>0$ and $2<x+y<3$ 
My approach:
\begin{align*}
1 &= \int_0^\infty\int_0^\infty cx\,dx\,dy
\\ &= \int_0^2\int_{2-x}^{3-x}cx \,dx\,dy
\end{align*}
I just want to know if the limits of the integrations are correct or not?

You just missed out that the upper limit on $x$ can be $3$ instead of $2$. Therefore the right integral is $\int_0^3\int_{2-x}^{3-x}cx\ dxdy$.
