What is the difference between a Limit and Derivative? 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.
 A: 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.
A: 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.
A: The idea of a limit is used to compute the derivative.
The derivative is really why Calculus was invented.  People wanted to say things about "instantaneous" rates of change, like what the velocity of a car is at a given point in time (which doesn't really make sense without the idea of a limit right?).  You are asking what the rate of change is at a single point, and change definitionally requires two points.
Thus, the idea of the limit was invented in order to calculate the derivative.  IE: the idea of assuming the value of a point given how a function behaves infinitely close to that point is required in order to say anything about the "rate of change" (slope of a tangent line) AT that point.
