- #1
mathshead
can someone explain what the formal difinition of a limit ?
The definition of a limit is a fundamental concept in mathematics and describes the behavior of a function as its input approaches a certain value. In other words, it is the value that a function approaches as its input gets closer and closer to a specific value.
A limit is typically written using the notation "lim f(x) = L" where f(x) is the function and L is the limit. The notation also includes the variable approaching the limit value. For example, "lim x → 3" means the limit of the function as x approaches 3.
A left-hand limit is the value that a function approaches as its input approaches a certain value from the left side. A right-hand limit, on the other hand, is the value that a function approaches as its input approaches a certain value from the right side. If the left-hand and right-hand limits are equal, then the overall limit exists.
If a limit does not exist, it means that either the left-hand and right-hand limits are not equal, or that one or both of the limits are infinite. In other words, the function does not approach a single, finite value as the input approaches the limit value.
The definition of a limit is a crucial concept in calculus and is used to determine the derivative and integral of a function. It is also used to analyze the behavior of functions and make predictions about their values. Without the concept of a limit, many fundamental principles in calculus would not be possible.