Limit of a function
For and an accumulation point of , is the limit of as when .
Let . Take such that is an accumulation point for . We say that as if
is called the limit of as , and we write .
In particular, for where is an accumulation point for , we say that diverges (to ) as if for some , and
Uniqueness
Suppose has a limit at , where is an accumulation point of . The limit is unique: if and as , then
The left side does not depend on while the right side does, so taking the limit gives , hence .
The definition only constrains points with , so it applies even when : what matters is that is an accumulation point of , so that points of the domain arbitrarily close to are available.
Example
has domain , and . A geometric argument on the trigonometric circle gives for all , hence
Given , choosing yields .
Related
Stated in
- Definition 2.4 (Limit of a Function)§2.1 Limits of Functions