I believed the definitions of derivative that we know was really definitions(adsbygoogle = window.adsbygoogle || []).push({});

[tex]f'(x_0)=\lim_{x\rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}[/tex]

[tex]f'(x_0)=\lim_{\Delta x\rightarrow 0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}[/tex]

But not, isdefinition, just use the equality bellow in equations above...one

[tex]\\ \Delta x=x-x_0 \\ x=x_0+\Delta x[/tex]

However, how proof the equality between limits?

[tex]\lim_{x\rightarrow x_0} \; \underset{=}{?} \; \lim_{\Delta x\rightarrow 0}[/tex]

Maybe by:

[tex]\lim_{x\rightarrow x_0}=\lim_{x-x_0\rightarrow x_0-x_0}=\lim_{\Delta x\rightarrow 0}[/tex]

What do you think about this? Is correct work with limit as if it is a equation?

EDIT: And more... Is possible to manipulate the limits like a algebric variable? I want say... move the limit from left side of equation to right side, apply a limit in equation that eliminate outher limit that already exist, know, use the limit like a function (and a function have a inverse function). What do you think??

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# Equations with limits

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