What is a Limit?
Limit: A limit describes the value that a function approaches as its input gets closer and closer to a particular value , regardless of what the function actually equals at that point.
Limits are the foundation of calculus. Before we can talk about derivatives, rates of change, or areas under curves, we need to understand what it means for a function to approach a value.
The formal notation for a limit is:
This reads: "The limit of f(x) as x approaches a equals L."
The key idea is that we are not asking what happens at , we are asking what value gets closer and closer to as gets closer and closer to .
Imagine you're walking toward a wall. A limit asks: "What point on the wall are you heading toward?" , not whether you actually reach it. You could stop just before the wall, or the wall might have a hole at that exact spot, but the destination is still clear from your direction of travel.