http://www.ck12.org Chapter 1. Functions, Limits, and Continuity
Since the domain off(x) =√xisx≥0, we see that that limx→ 0 √xdoes not exist. Specifically, we cannot find
open intervals aroundx=0 that satisfy the limit definition. However we do note that as we approachx=0 from the
right-hand side, we see the successive values tending towardsx=0. This example provides some rationale for how
we can defineone-sided limits.
Definition:
We say that theright-handlimitof a functionf(x)ataisb, written as limx→a+f(x) =b, if for every open interval
Nofb, there exists an open interval(a,a+δ)contained in the domain off(x),such thatf(x)is inNfor everyxin
(a,a+δ).
For the example above, we write limx→ 0 +√x= 0.
Similarly, we say that theleft-hand limitoff(x)ataisb, written as limx→a−f(x) =b, if for every open interval
Nofbthere exists an open interval(a−δ,a)contained in the domain off(x),such thatf(x)is inNfor everyxin
(a−δ,a).
Example 1:
Find limx→ 0 +|xx|.
The graph has a discontinuity atx=0 as indicated: