Why doesn't my derivation of a product work?

[georg gill]

What do I do wrong here:\frac{f(x+h)g(x+h)-f(x)g(x)}{x+h-x}=\frac{f(x+h)g(x+h)}{h}-\frac{f(x)g(x)}{h}=\frac{f(x+h)}{h}g(x+h)-\frac{f(x)}{h}g(x)=\lim_{h \to 0}\frac{f(x+h)}{h}\lim_{h \to 0}g(x+h)-\lim_{h \to 0}\frac{f(x)}{h}g(x)\lim_{h \to 0}g(x+h)=g(x)
\lim_{h \to 0}\frac{f(x+h)}{h}\lim_{h \to 0}g(x+h)-\lim_{h \to 0}\frac{f(x)}{h}g(x)=\lim_{h \to 0}\frac{f(x+h)}{h}g(x)-\lim_{h \to 0}\frac{f(x)}{h}g(x)=(\lim_{h \to 0}\frac{f(x+h)}{h}-\lim_{h \to 0}\frac{f(x)}{h})g(x)=(\lim_{h \to 0}\frac{f(x+h)-f(x)}{h})g(x)=\frac{df}{dx}g(x)I know how they derieve the derivation of a product:

http://bildr.no/view/918745

But how come what I did above does not work?

 

[micromass]

You did something like

\lim_{x\rightarrow a}{f(x)g(x)+h(x)}=\lim_{x\rightarrow a}{f(x)}\lim_{x\rightarrow a}{g(x)}+\lim_{x\rightarrow a}{h(x)}

But this is not true. You can't do that.

It is ONLY true if \lim_{x\rightarrow a}{f(x)}, \lim_{x\rightarrow a}{g(x)} AND \lim_{x\rightarrow a}{h(x)} converge. In your example, you don't have that. For example

\lim_{h\rightarrow 0}{\frac{f(x)}{h}}

does not converge.