Derivation af oa product

  • Thread starter georg gill
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  • #1
georg gill
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Main Question or Discussion Point

What do I do wrong here:


[tex]\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)[/tex]


[tex]\lim_{h \to 0}g(x+h)=g(x)[/tex]



[tex]\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)[/tex]


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?
 
Last edited:

Answers and Replies

  • #2
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You did something like

[tex]\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)}[/tex]

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

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

[tex]\lim_{h\rightarrow 0}{\frac{f(x)}{h}}[/tex]

does not converge.
 

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