Limits are really important for understanding how functions work. They help us connect different ideas in math. Let's break it down into simpler parts:

1. What is Continuity?
   A function is called continuous at a point ( c ) if it meets three rules:

   • The function ( f(c) ) is defined, which means you can find a value at that point.
   • The limit of ( f(x) ) as ( x ) gets closer to ( c ) exists, meaning we can see what value ( f(x) ) approaches.
   • The limit must equal the function's value at that point, written as ( \lim_{x \to c} f(x) = f(c) ).

2. How Limits Help
   Limits show us what happens to the function as we get really close to ( c ). If both the left-hand limit (approaching from the left) and the right-hand limit (approaching from the right) give us the same number, it means there is no sudden jump or break in the graph.

3. Finding Discontinuities
   If any of the three rules for continuity aren’t met, it means the function has a discontinuity, or a break. For example:

   • Jump discontinuities happen when the left and right limits give different results.
   • Infinite discontinuities occur when the function goes off to infinity.

4. Real-Life Examples
   I remember working with piecewise functions, which are functions made up of different pieces. You have to check if those pieces connect smoothly at certain points. It was surprising because even if a function seems to be defined everywhere, limits can show us hidden breaks we didn’t notice before.

In short, limits give us important information about how functions behave. They are necessary for understanding continuity and discontinuity in algebra.