section_path: Types of limits/In functions/One-sided limit
section_name: One-sided limit
paragraph_text: One-sided limit
  
  It is possible to define the notion of having a "left-handed" limit ("from below"), and a notion of a "right-handed" limit ("from above"). These need not agree. An example is given by the positive indicator function, f: R → R {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }, defined such that f (x) = 0 {\displaystyle f(x)=0} if x ≤ 0 {\displaystyle x\leq 0}, and f (x) = 1 {\displaystyle f(x)=1} if x > 0 {\displaystyle x>0}. At x = 0 {\displaystyle x=0}, the function has a "left-handed limit" of 0, a "right-handed limit" of 1, and its limit does not exist. Symbolically, this can be stated as, for this example, lim x → c − f (x) = 0 {\displaystyle \lim _{x\to c^{-}}f(x)=0}, and lim x → c + f (x) = 1 {\displaystyle \lim _{x\to c^{+}}f(x)=1}, and from this it can be deduced lim x → c f (x) {\displaystyle \lim _{x\to c}f(x)} doesn't exist, because lim x → c − f (x) ≠ lim x → c + f (x) {\displaystyle \lim _{x\to c^{-}}f(x)\neq \lim _{x\to c^{+}}f(x)}.