A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It represents the value that the function is approaching, rather than the actual value at that point.
A limit is calculated by evaluating the function at values that are increasingly closer to the desired input value. This process is known as taking the limit. If the function approaches a specific value as the input gets closer and closer, then that value is the limit.
A supremum, or least upper bound, is the smallest possible number that is greater than or equal to all the numbers in a set. In other words, it is the smallest upper bound of a set of numbers.
The supremum of a set is determined by finding the largest number in the set. If there is no largest number, then the supremum does not exist. In some cases, the supremum can be infinite.
Limits and supremum are related in that the limit of a function can be equal to the supremum of a set. This occurs when the function is continuous and the supremum of the set is within the function's range. However, in most cases, the limit and supremum are two distinct concepts with different ways of being calculated.