Question about the derivative definition The derivative at a point $x$ is defined as:
$\lim\limits_{h\to0} \frac{f(x+h) - f(x)}h$
But if $h\to0$, wouldn't that mean:
$\frac{f(x+0) - f(x)}0 = \frac0{0}$ which is undefined?

Actually, the $ \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $ is the value which $\frac{f(x+h) - f(x)}{h}$ approaches when you keep reducing $h$ as much as you can.
Here is how my teacher explained me the idea:

Think of drawing a tangent. It's a line touching a curve at only one point.
But how could that be possible? Two points determine a line. 
The tangent is the limiting case of a secant where the two points close in on each other.

Link to Animation
Like he said, and you correctly pointed out, the expression is meaningless if $h=0$.
Here is another paradox that Zeno used:

Suppose an Arrow is shot which travels from A to B.
Consider any instant.
Since, no time elapses during the instant, the arrow does not move
  during the motion.
But the entire time of flight consists of instances alone.
Hence, the arrow must not have moved.

The key here is to introduce the idea of speed, which is once again, a limit, the distance travelled per duration as the duration approaches an instant (tends to zero).
