Definition of Limit: A limit is a specific value that a function approaches as the input approaches a specific value. In other words, it represents the valu...
Definition of Limit:
A limit is a specific value that a function approaches as the input approaches a specific value. In other words, it represents the value that the function would take if it were evaluated at that input value.
Formal Definition:
Let (f) be a function defined on an open interval ((a, b)) containing the point (c). The limit of (f) as (x) approaches (c) is denoted by (\lim_{x\to c} f(x)) or (L\lim_{x\to c} f(x)), where (L) is the limit itself.
Sequential Criterion:
A function (f) has a limit (L) at (c) if the sequence (f(x)) converges to (L) as (x) approaches (c), regardless of the path the sequence takes. In other words, ( \lim_{x\to c} f(x) = L) if the sequence (f(x)) approaches (L) regardless of whether we take the path from the left or the right of (c).
Examples:
( \lim_{x\to 0} \frac{1}{x} = 1)
( \lim_{x\to \infty} x^2 = \infty)
( \lim_{x\to 1} (x-1) = 0)
By definition, the limit of a function can be either a finite real number or an infinite value, including infinity and negative infinity