What is the Point of Limits?

What is the point of limits? Not the english word but the mathematical definition? It's the first topic you learn in Calculus 1, and is the foundation for the rest of the calculus series. All of calculus rests on the idea of limits, but why is that? It’s because calculus studies change and accumulation, and to get to a good approximation getting as close as possible to a given constraint is critical, which is what a limit is.

Starting with derivatives, derivatives measure the change at an exact time, often referenced as instantaneous change. The equation for the derivative is:

This definition is basically saying the change between two points, but make the change between the points so small it might as well be the change only at one point.

Limits come into integrals as well as seen in the equation:

Now this looks a  little more complicated, but its idea is simple. Basically taking a shape, and then cutting the shape into infinity pieces to then evaluate the size of each individual piece, and then add them up to find the total area of the object.

The rest of the calculus series relies on the idea of derivatives and integrals, which each do have easier methods than the two shown above. The power rule, and then many more but even with these other methods none of it would have been created without limits. So in a sense, calculus is the study of the limit.