I want to know whether I can change the order of integral and limit, as following $$ \lim_\limits{\Delta x\rightarrow0}\frac{ \int_{a}^{b}y\left[ H(z+\frac{\Delta x}{y}) -H(z) \right]f(y)dy }{ \Delta x } =\int_a^b\left[ \lim_\limits{\Delta x\rightarrow0} \frac{ H(z+\frac{\Delta x}{y}) -H(z) }{ \frac{\Delta x}{y} } \right]f(y)dy \\ = \int_a^b H'(z) f(y)dy $$
Or under which condition I can do this?