In the previous lesson, we were concerned with the values of x \neq a when considering the limit of the function f as it approaches a in the open interval I that contains a (what a mouthful). For those limits, we weren't sure of the direction approaching a (be it the left side of the limit or the right side). We just considered both values of x coming from the left and right of a. This is the standard definition of a twosided limit. But as you can guess (from the title of this page I hope), there are also onesided limits that will be addressed here.
Onesided limits are important when there is a discontinuity of the function on either side of a. When that situation occurs, a twosided limit will not work. It may be true on one side but it can't be on the other side. An example would do a better job of explaining this: consider the function f(x)=\sqrt{1+x} which is plotted in Figure 1.
Note that the value of the function is imaginary when x<1. This corresponds to the left side of the the limit supposing that a=1. Hence, f(x) is not defined on the left side of a. So, a two sided limit would not be applicable to this type of problem. However, if we consider the values of x that are greater than 1, we find that the values of f are defined! See! So this means that on one side, the limit \lim_{x \to 1^{+}}\sqrt{1+x} exists. Thus, it is important to take time studying not just the twosided limits but also onesided limits as well which come in two flavors.
The Two Types of OneSided Limits  
As mentioned, there are two types of onesided limits. The first type is when the value of x approaches from the right of a (similar to the example we showed you above). This type of limit would only consider the values of x to the right of a. In limit notation, this goes like,
\lim_{x \to a^{+}} f(x)=L
which is read as "L is the limit of f(x) as x approaches a from the right". The symbol "x \to a^{+}" means that x approaches a through values greater than a. Knowing this, we can now define a righthanded limit.
Let f be a function defined at every point in some open interval (a,b). Then "the limit of f(x) as x approaches a from the right is equal to L", notated as:
\lim_{x \to a^{+}} f(x)=L

For every (\epsilon>0) (no matter how small) there will exist a (\delta>0) such that f(x)L< \epsilon whenever [0< xa <\delta] .
Likewise, "L can be the limit of f(x) as x approaches a from the left", and we'll write this as:
\lim_{x \to a^{}} f(x)=L
The symbol "x \to a^{}" means that x approaches a through values less than a. With all these in mind, let us now define leftsided limits formally:
Let f be a function defined at every point in some open interval (c,a). Then "the limit of f(x) as x approaches a from the left is equal to L", notated as
\lim_{x \to a^{+}} f(x)=L

For every (\epsilon >0) (no matter how small) there will exist a (\delta>0) such that f(x)L< \epsilon whenever [0< xa <\delta] .
As an important remark, the theorems discussed in the previous article still holds true for onesided limits where "xa" is replaced by "xa^{+}" or "xa^{}".
Evaluate the following limit:
Solution: Notice that if we directly follow theorems discussed before by substituting 1 to the function, it gives zero at the denominator. To avoid these, let us rationalize the denominator. Note that rationalizing the denominator is not a standard option in solving onesided limits. The use of this method is on a case to case basis. Well anyway, let's proceed.
First, let us rationalize the function
f(x)  =  \frac{x1}{\sqrt{x^{2}1}} 
=  \frac{x1}{\sqrt{ (x+1)(x1)}} * {\sqrt{ (x1) (x+1)}\over {\sqrt{ (x+1) (x1) }}}  
=  \frac{(x1)\sqrt{x^{2}1}}{\sqrt{ (x+1)^{2}(x1)^{2}}}  
=  \frac{(x1)\sqrt{x^{2}1}}{ (x+1)(x1) }  
=  \frac{\sqrt{x^{2}1}}{ (x+1) } 
We have now succesfully rationalized the denominator. Observe that it is not zero at the denominator when x is replaced by a.
Thus, we now proceed accordingly.
\lim_{x \to 1^{+}} f(x)  =  \lim_{x \to 1^{+}} \frac{\sqrt{x^{2}1}}\over{ (x+1) } 
=  \frac{\sqrt{1^{2}1}}{1+1}  
=  \frac{0}{2}  
=  0 
In a broader sense, the solutions and techniques we use to solve twosided limits are still applicable for onesided limits, as well as the theorems involved.
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