When you get the derivative of a point, arn't you just getting the limit at that point?
I'm not quite sure why they need to be named differently when they seem to be doing the same thing.
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$\begingroup$Limit is a tool which we used to compute the derivative.
For example, $$f'(x_0)=\lim_{x\to 0 }\frac{f(x_0+x)-f(x_0)}{x}.$$
We use Limit to get derivative.
$\endgroup$ $\begingroup$Here's an example, let's say that $$ f(x)=x^2 $$ By finding the limit of $f(x)$, we can see the behavior of $f(x)$ as $f(x)$ approaches $c$. So if $c=0$, then $$ \lim_{x\to 0} f(x) = \lim_{x\to 0} x^2 = 0^2 = 0 $$ The derivative of $f(x)$ is a specific type of limit and is defined as $$ \frac{d}{dx}f(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h} $$ So now let's find the derivative of $f(x)$ by using the limit definition $$\frac{d}{dx}x^2= \lim_{h\to 0} \frac{(x+h)^2-x^2}{h} $$ $$ = \lim_{h\to 0} \frac{x^2+2xh+h^2-x^2}{h}= \lim_{h\to 0} \frac{2xh+h^2}{h} $$ $$ = \lim_{h\to 0} [2x+h]=2x+0=2x $$ The concept of the limit is more general than the concept of the derivative.
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