Single-variable calculus from first principles
A limit answers a careful question: as the input gets closer and closer to some value a, what number does the output home in on? Crucially, it doesn't matter what happens at a; maybe the function isn't even defined there. The limit is about the approach, not the destination itself.
Drag the input toward a in the figure and watch the output settle onto a value L, even across a little hole where the function has no value of its own.
You can approach a from the left (inputs a little below a) or from the right (a little above). These are the two one-sided limits. The full (two-sided) limit exists only when both sides agree on the same number. If the left side heads to one value and the right to another, there's a jump, and the limit doesn't exist.