Can $\lim_{h\to0}\frac{f(h+x_0)-f(x_0)}{h}$ be written as $\lim_{h\to0}\frac{f(x)-f(x-h)}{h}$ when $h=x−x_0$?

Question

Say that we have a function $f(x)=x^2$, and two distinct points $P(x_0,f(x_0))$ and $Q(x,f(x))$ on the curve $y=f(x)$; if we let $Q$ approach $P$, then the slope of the secant through $Q$ and $P$ will approach a limit, which is the slope of the tangent line at $P$:$$m_{tan}= \lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}. . . . .(1)$$When $h=x-x_0$, then:$$m_{tan}= \lim_{h\to 0}\frac{f(x_0+h)-f(x_0)}{h}. . . . .(2)$$I have tried to write $(1)$ as:$$\lim_{h\to0}\frac{f(x)-f(x-h)}{h}. . . . .(3)$$ instead of $(2)$, but when I tried finding the limit I got a different answer than that of $(2)$.
So, what is the point here?