A limit of a function is the value that a function approaches as the input approaches a certain value. It is not necessarily the value that the function outputs, but rather the value that it gets closer and closer to as the input gets closer to a specific value.
The limit of a function can be calculated by evaluating the function at values that are very close to the desired input value. This can be done using a table, graph, or algebraic methods such as factoring and simplifying the function.
Limits are important in calculus because they allow us to understand the behavior of a function at a specific point. They also help us determine the continuity and differentiability of a function, which are fundamental concepts in calculus.
A one-sided limit only considers the behavior of a function as the input approaches a specific value from one side (either the left or right). A two-sided limit, on the other hand, considers the behavior of a function as the input approaches the specific value from both sides.
A limit does not exist if the function has a jump discontinuity or an infinite discontinuity at the specific value. This means that the function does not approach a single, finite value as the input gets closer to the specific value.