How are limits and derivatives related?

Question

irishboy123 · Answer

limits allow you to do some very naughty albegra, specifically retriceociating a value with $\dfrac{0}{0}$!

gonzo2299 · Answer

Okay, but how does that relate limits to derivatives?

irishboy123 · Answer

we can do an example

gonzo2299 · Answer

Okay
do show

irishboy123 · Answer

$f(x) = x^2$

by definition

$f'(x) =\lim\limits_{h \to 0} \dfrac{(x+h)^2 - x^2}{h}$

irishboy123 · Answer

$ =\lim\limits_{h \to 0} \dfrac{x^2 + 2 hx + h^2- x^2}{h}$

$ =\lim\limits_{h \to 0} \dfrac{ 2 hx + h^2}{h}$

gonzo2299 · Answer

Ok, so they relate because the limit is needed to find the derivative?

irishboy123 · Answer

yes!

look at the example, we currently have h in numerator and denominator.  $\dfrac{0}{0}$

limits allow us to deal with that

gonzo2299 · Answer

Alright sweet!

thanks for helping clear that up

irishboy123 · Answer

ouch ouch ouch, just to be clear $h \ne 0$ :-)  the limits see to that

mathmale · Answer

Note that all derivative formulas are the end results of having taken a limit of one kind or another.

So limits are the basis on which derivatives are built.