Subject: Calculus

# One Sided Limits

## Calculus.One-sidedLimit History

June 26, 2011 by matthew_suan -
Changed line 73 from:
!!Example #1
to:
!!!Example #1

'''NEXT TOPIC:'''[[Limit of a sequence]]
December 31, 2010 by matthew_suan -
December 31, 2010 by matthew_suan -
November 01, 2010 by misslee -
Changed lines 44-46 from:
For every {$\ (epsilon >0)$} (no matter how small) there will exist a {$(\delta>0)$} such that {$|f(x)-L|< \epsilon$} whenever {$[0< x-a <\delta]$}.
to:
For every {$(\epsilon>0)$} (no matter how small) there will exist a {$(\delta>0)$} such that {$|f(x)-L|< \epsilon$} whenever {$[0< x-a <\delta]$}.
Changed line 66 from:
For every {$\ (epsilon >0)$} (no matter how small) there will exist a {$(\delta>0)$} such that {$|f(x)-L|< \epsilon$} whenever {$[0< x-a <\delta]$}.
to:
For every {$(\epsilon >0)$} (no matter how small) there will exist a {$(\delta>0)$} such that {$|f(x)-L|< \epsilon$} whenever {$[0< x-a <\delta]$}.
November 01, 2010 by misslee -
Changed lines 22-23 from:
(:tableend:)
to:
(:tableend:)\\
November 01, 2010 by misslee -
Changed lines 2-3 from:
In the [[limit of a function | 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 two-sided limit. But as you can guess (from the title of this page I hope), there are also one-sided limits that will be addressed here.\\
to:
In the [[limit of a function | 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 two-sided limit. But as you can guess (from the title of this page I hope), there are also one-sided limits that will be addressed here.\\\
Changed lines 7-10 from:
One-sided limits are important when there is a discontinuity of the function on either side of {$a$}. When that situation occurs, a two-sided 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 two-sided limits but also one-sided limits as well which come in two flavors. \\
to:
One-sided limits are important when there is a discontinuity of the function on either side of {$a$}. When that situation occurs, a two-sided 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 two-sided limits but also one-sided limits as well which come in two flavors. \\\
November 01, 2010 by misslee -
Changed line 2 from:
In the [[limit of a function | 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 sureof 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 two-sided limit. But as you can guess (from the title of this page I hope), there are also one-sided limits that will be addressed here.\\
to:
In the [[limit of a function | 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 two-sided limit. But as you can guess (from the title of this page I hope), there are also one-sided limits that will be addressed here.\\
November 01, 2010 by misslee -
Changed lines 2-3 from:
In the [[limit of a function | 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 had no predilection for which direction approached {$a$} from (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 two-sided limit. But as you can guess (from the title of this page I hope), there are also one-sided limits that will be addressed here.\\
to:
In the [[limit of a function | 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 sureof 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 two-sided limit. But as you can guess (from the title of this page I hope), there are also one-sided limits that will be addressed here.\\
Changed lines 6-11 from:
One-sided limits are important when there is a discontinuity of the function on either side of {$a$}. When that occurs a two-sided 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 two-sided limits but also one-sided limits as well which com in two flavors. \\
to:
One-sided limits are important when there is a discontinuity of the function on either side of {$a$}. When that situation occurs, a two-sided 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 two-sided limits but also one-sided limits as well which come in two flavors. \\
Changed lines 30-31 from:
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 right-handed limit
to:
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 right-handed limit.
Changed line 77 from:
''Solution:'' Notice that if we follow directly theorems discussed before by substituting directly 1 to the function, it gives zero at the denominator. To avoid these, let us rationalize the denominator. Note that rationalizing denominator is not a standard option in solving one-sided limits. The use of such is on a case to case basis. Well anyway, let's proceed.\\
to:
''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 one-sided limits. The use of this method is on a case to case basis. Well anyway, let's proceed.\\
Changed line 100 from:
In a broader sense, the solutions and techniques employed in solving two-sided limits are still applicable for one-sided limits, as well as the [[Limit of a function | theorems]] involved.
to:
In a broader sense, the solutions and techniques we use to solve two-sided limits are still applicable for one-sided limits, as well as the [[Limit of a function | theorems]] involved.
October 27, 2010 by matthew_suan -
October 27, 2010 by math2 -
Changed line 82 from:
|| || = ||'+{${\frac{x-1}{\sqrt{ (x+1)(x-1) }}} x {\frac{\sqrt{ (x-1) (x+1) }}{\sqrt{ (x+1) (x-1) }}}$}+' ||
to:
|| || = ||'+{$\frac{x-1}{\sqrt{ (x+1)(x-1)}} *$} {${\sqrt{ (x-1) (x+1)}\over {\sqrt{ (x+1) (x-1) }}}$}+' ||
Changed line 93 from:
|| '+{$\lim_{x \to 1^{+}} f(x)$}+'|| = ||'+{$\lim_{x \to 1^{+}} \frac{\sqrt{x^{2}-1}}{ (x+1) }$}+' ||
to:
|| '+{$\lim_{x \to 1^{+}} f(x)$}+'|| = ||'+{$\lim_{x \to 1^{+}} \frac{\sqrt{x^{2}-1}}\over{ (x+1) }$}+' ||
October 26, 2010 by math2 -
Changed line 82 from:
|| || = ||'+{$\frac{x-1}{\sqrt{ (x+1)(x-1) }} x \frac{\sqrt{ (x-1) (x+1) }}{\sqrt{ (x+1) (x-1) }}$}+' ||
to:
|| || = ||'+{${\frac{x-1}{\sqrt{ (x+1)(x-1) }}} x {\frac{\sqrt{ (x-1) (x+1) }}{\sqrt{ (x+1) (x-1) }}}$}+' ||
October 26, 2010 by math2 -
Changed lines 2-15 from:
In the previous lessons, we are concerned with the values of {$x \neq a$} when considering the limit of {$f$} as it approaches {$a$} in an open interval {$I$} that are close and yet, contains {$a$}. We did work out on those limits without being particular to which direction the approach is coming. We just consider both values of {$x$} coming from the left and right of {$a$}. This kind of mechanism is attributed to two-sided limits. \\
[[<<]]
!!Why
is there a single-sided limits?

%rframe
width=300px border='0px solid black' padding=2px%Attach:one_sided_limits.jpg | '''Figure 1: Plot of the function {$f(x)$}. [[http://www.mathematics2.com/Calculus/One-sidedLimit?action=download&upname=one_sided_limits.jpg | (Source)]]'''

One
sided limits are considered and is important because when there is a discontinuity of the function on either side of {$a$}, then the two-sided limits is not true at all. It may be true on one side but it can't be on the other side. As an illustration, consider the function {$f(x)=\sqrt{1+x}$} which is plotted in Figure 1.\\

Note
that the value of the function is imaginary at points {$x<-1$} as seen in the figure. 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 {$a$}, thus, the two sided limit does not makes sense in this type of configuration. However, if we consider the values of {$x$} that are greater than -1, then 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 two-sided limits but also one-sided limits as well which comes in two types. \\
[[<<]]
!!Classifications of One
-Sided Limits
to:
In the [[limit of a function | 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 had no predilection for which direction approached {$a$} from (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 two-sided limit. But as you can guess (from the title of this page I hope), there are also one-sided limits that will be addressed here.\\

!!You
lost me. Why would there be a single-sided limit?
%rframe
width=300px border='0px solid black' padding=2px%Attach:one_sided_limits.jpg | ''''-Figure 1: Plot of the function {$f(x)$}. [[http://www.mathematics2.com/Calculus/One-sidedLimit?action=download&upname=one_sided_limits.jpg | (Source)]]-''''
One-sided
limits are important when there is a discontinuity of the function on either side of {$a$}. When that occurs a two-sided 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 two-sided limits but also one-sided limits as well which com in two flavors. \\
Changed line 13 from:
(:cellnr bgcolor=#d4d7ba colspan=14 align=center:) '+'''Two Types of One-Sided Limits'''+'
to:
(:cellnr bgcolor=#d4d7ba colspan=14 align=center:) '+'''The Two Types of One-Sided Limits'''+'
Changed line 16 from:
*[[#t2 | Right-Hand Limit: Definition]]
to:
**[[#t2 | Right-Hand Limit: Definition]]
Changed lines 18-19 from:
*[[#t4 | Left-Hand Limit: Definition]]
to:
**[[#t4 | Left-Hand Limit: Definition]]
* [[#ex1 | An example of a one-sided limit
]]
Changed lines 24-25 from:
As mentioned, there are two types of one-sided limits. The first type is when the value of {$x$} approaches from the right of {$a$}. This is so because we are considering the values of {$x$} to the right of {$a$}. This only happens when {$f$} is properly defined on that side. In limit notation, this goes like,
to:
As mentioned, there are two types of one-sided 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,
Changed lines 27-31 from:
\\\

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 {$a$}. Having known this one, let us know define right-handed limit formally.
to:
\\

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 right-handed limit
Changed lines 33-35 from:
!!Right-Hand Side Limit: Definition
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 {$L$}, notated as
to:
!!!Right-Hand Side Limit: Definition
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:
Changed lines 41-43 from:
if for every {$\epsilon >0$}, no matter how small, there exists a {$\delta>0$} such that {$|f(x)-L|< \epsilon$} whenever {$0<x-a<\delta$}.
to:
For every {$\ (epsilon >0)$} (no matter how small) there will exist a {$(\delta>0)$} such that {$|f(x)-L|< \epsilon$} whenever {$[0< x-a <\delta]$}.
Changed lines 46-47 from:
Likewise, when "L is the limit of {$f(x)$} as {$x$} approaches a from the left, then we may write it as
to:
Likewise, "L can be the limit of {$f(x)$} as {$x$} approaches a from the left", and we'll write this as:
Changed lines 49-53 from:
\\\

The symbol "{$x \to a^{-}$}" means that {$x$} approaches {$a$} through values less than {$a$}. With all these in mind, let us now define left-sided limits formally.
to:
\\

The symbol "{$x \to a^{-}$}" means that {$x$} approaches {$a$} through values less than {$a$}. With all these in mind, let us now define left-sided limits formally:
Changed lines 55-58 from:
!!Left-Hand Side Limit: Definition
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 {$L$}, notated as
to:
!!!Left-Hand Side Limit: Definition
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
Changed lines 63-67 from:

if for every {$\epsilon >0$}, no matter how small, there exists a {$\delta>0$} such that {$|f(x)-L|< \epsilon$} whenever {$0<x-a<\delta$}.
to:
For every {$\ (epsilon >0)$} (no matter how small) there will exist a {$(\delta>0)$} such that {$|f(x)-L|< \epsilon$} whenever {$[0< x-a <\delta]$}.
Changed lines 67-69 from:
!!!Example #1
Evaluate the following limits:
to:
\\\

[[#ex1]]
!!Example #1
Evaluate the following limit:
Changed line 82 from:
|| || = ||'+{$\frac{x-1}{\sqrt{ (x+1)(x-1) }} \cdot\frac{\sqrt{ (x-1) (x+1) }}{\sqrt{ (x+1) (x-1) }}$}+' ||
to:
|| || = ||'+{$\frac{x-1}{\sqrt{ (x+1)(x-1) }} x \frac{\sqrt{ (x-1) (x+1) }}{\sqrt{ (x+1) (x-1) }}$}+' ||
Changed line 100 from:
In a broader sense, the solutions and techniques employed in solving two-sided limits are still applicable for one-sided limits, as well as the theorems involved.
to:
In a broader sense, the solutions and techniques employed in solving two-sided limits are still applicable for one-sided limits, as well as the [[Limit of a function | theorems]] involved.
October 26, 2010 by math2 -
Changed line 2 from:
In the previous lessons, we are concerned with the values of {$x \neq a$} when considering the limit of {$f$} as it approaches {$a$} in an open interval {$I$} that are close and yet, contains {$a$}. We did work out on those limits without being particular to which direction the approach is coming. We just consider both values of {$x$} coming from the left and right of {$a$}. This kind of mechanism is attributed to two-sided limits. \\
to:
In the previous lessons, we are concerned with the values of {$x \neq a$} when considering the limit of {$f$} as it approaches {$a$} in an open interval {$I$} that are close and yet, contains {$a$}. We did work out on those limits without being particular to which direction the approach is coming. We just consider both values of {$x$} coming from the left and right of {$a$}. This kind of mechanism is attributed to two-sided limits. \\
October 26, 2010 by math2 -
Changed line 1 from:
(:title: One Sided Limits:)
to:
(:title One Sided Limits:)
October 26, 2010 by math2 -
(:title: One Sided Limits:)
October 26, 2010 by matthew_suan -
Changed line 2 from:
to:
[[<<]]
October 26, 2010 by matthew_suan -
Changed line 73 from:
\begin{example}
to:
!!!Example #1
Changed lines 76-108 from:
\begin{equation}
\lim_{x \to 1^{+}} \frac{x-1}{\sqrt{x^{2}-1}}
\end{equation}

\end{example}

\indent
Notice
that if we follow directly theorems discussed before by substituting directly 1 to the function, it gives zero at the denominator. To avoid these, let us rationalize the denominator. Note that rationalizing denominator is not a standard option in solving one-sided limits. The use of such is on a case to case basis. Well anyway, let's proceed.
\newline{}
\indent
First,
let us rationalize the function

\begin{eqnarray}
f(x) & = & \frac{x-1}{\sqrt{x^{2}-1}} \\
& = & \frac{x-1}{\sqrt{(x+1)(x-1)}}\cdot \frac{\sqrt{(x-1)(x+1)}}{\sqrt{(x+1)(x-1)}} \\
& = & \frac{(x-1)\sqrt{x^{2}-1}}{\sqrt{(x+1)^{2}(x-1)^{2}}} \\
& = & \frac{(x-1)\sqrt{x^{2}-1}}{(x+1)(x-1)} \\
& = & \frac{\sqrt{x^{2}-1}}{(x+1)}
\end{eqnarray}

\noindent
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.

\begin{eqnarray}
\lim_{x \to 1^{+}} f(x) & = & \lim_{x \to 1^{+}} \frac{\sqrt{x^{2}-1}}{(x+1)} \\
& = & \frac{\sqrt{1^{2}-1}}{1+1} \\
& = & \frac{0}{2} \\
& = & 0
\end
{eqnarray}

Thus,
we have successfully solved for the limit of $f(x)$. In a broader sense, the solutions and techniques employed in solving two-sided limits are still applicable for one-sided limits, as well as the theorems involved.

\end{document}
to:
'+{$$\lim_{x \to 1^{+}} \frac{x-1}{\sqrt{x^{2}-1}}$$}+'

(:toggle
id=box1 show='Show Answers with Solutions' init=hide button=1:)
>>id=box1
border='1px solid #999' padding=5px bgcolor=#edf<<
''Solution:''
Notice that if we follow directly theorems discussed before by substituting directly 1 to the function, it gives zero at the denominator. To avoid these, let us rationalize the denominator. Note that rationalizing denominator is not a standard option in solving one-sided limits. The use of such is on a case to case basis. Well anyway, let's proceed.\\
First, let us rationalize the function

||
border=0 align=center width=80%
|| '+
{$f(x)$}+'|| = ||'+{$\frac{x-1}{\sqrt{x^{2}-1}}$}+' ||
||
|| = ||'+{$\frac{x-1}{\sqrt{ (x+1)(x-1) }} \cdot\frac{\sqrt{ (x-1) (x+1) }}{\sqrt{ (x+1) (x-1) }}$}+' ||
||
|| = ||'+{$\frac{(x-1)\sqrt{x^{2}-1}}{\sqrt{ (x+1)^{2}(x-1)^{2}}}$}+' ||
||
|| = ||'+{$\frac{(x-1)\sqrt{x^{2}-1}}{ (x+1)(x-1) }$}+' ||
||
|| = ||'+{$\frac{\sqrt{x^{2}-1}}{ (x+1) }$}+' ||

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.

||
border=0 align=center width=80%
||
'+{$\lim_{x \to 1^{+}} f(x)$}+'|| = ||'+{$\lim_{x \to 1^{+}} \frac{\sqrt{x^{2}-1}}{ (x+1) }$}+' ||
||
|| = ||'+{$\frac{\sqrt{1^{2}-1}}{1+1}$}+' ||
||
|| = ||'+{$\frac{0}{2}$}+' ||
||
|| = ||'+0+' ||

>><<

In a broader sense, the solutions and techniques employed in solving two-sided limits are still applicable for one-sided limits, as well as the theorems involved.
October 26, 2010 by math2 -
Changed line 5 from:
%rframe width=200px border='0px solid black' padding=2px%Attach:one_sided_limits.jpg | '''Figure 1: Plot of the function {$f(x)$}.'''
to:
%rframe width=300px border='0px solid black' padding=2px%Attach:one_sided_limits.jpg | '''Figure 1: Plot of the function {$f(x)$}. [[http://www.mathematics2.com/Calculus/One-sidedLimit?action=download&upname=one_sided_limits.jpg | (Source)]]'''
October 26, 2010 by matthew_suan -
Changed lines 19-21 from:
*[[#t2 | Left-Handed Limit]]
to:
*[[#t2 | Right-Hand Limit: Definition]]
*[[#t3 | Left-Handed Limit]]
*[[#t4 | Left-Hand Limit: Definition
]]
[[#t1]]
!! Right-Handed Limit
Changed lines 28-29 from:
to:
%cframe align=center%'+{$\lim_{x \to a^{+}} f(x)=L$}+'
\\\

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 {$a$}. Having known this one, let us know define right-handed limit formally.

[[#t2]]
!!Right-Hand Side Limit: Definition
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 {$L$}, notated as
Changed lines 43-45 from:
which is read as "L is the limit of {$f(x)$} as {$x$} approaches a from the right". Likewise, when "L is the limit of {$f(x)$} as {$x$} approaches a from the left, then we may write it as
to:
if for every {$\epsilon >0$}, no matter how small, there exists a {$\delta>0$} such that {$|f(x)-L|< \epsilon$} whenever {$0<x-a<\delta$}.

[[#t3]]
!!Left-Handed
Limit
Likewise, when "L is the limit of {$f(x)$} as {$x$} approaches a from the left, then we may write it as

%cframe align=center%'+{$\lim_{x \to a^{-}} f(x)=L$}+'
\\\

The symbol "{$x \to a^{-}$}" means that {$x$} approaches {$a$} through values less than {$a$}. With all these in mind, let us now define left-sided limits formally.

[[#t4]]
!!Left-Hand Side Limit: Definition
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 {$L$}, notated as
Changed line 63 from:
'+{$$\lim_{x \to a^{-}} f(x)=L$$}+'
to:
'+{$$\lim_{x \to a^{+}} f(x)=L$$}+'
Changed lines 67-89 from:
The symbol "{$x \to a^{+}$}" means that {$x$} approaches {$a$} through values greater than {$a$} while the symbol "{$x \to a^{-}$}" means that {$x$} approaches {$a$} through values less than {$a$}. With all these in mind, let us now define those one-sided limits formaly.

[[#t1]]
!!Right-Hand
Side 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 {$L$}, notated as

\begin{equation}
\lim_{x \to a^{+}} f(x)=L
\end{equation}
if for every $\epsilon >0$, no matter how small, there exists a $\delta>0$ such that $|f(x)-L|< \epsilon$ whenever $0<x-a<\delta$.
\end{definition}

\begin{definition}[Left-Hand Side Limit]
Let $f$ be a function defined at every point in some open interval (c,a). Then \textbf{the limit of f(x) as x approaches a from the left is L}, notated as

\begin{equation}
\lim_{x \to a^{-}} f(x)=L
\end{equation}
if for every $\epsilon >0$, no matter how small, there exists a $\delta>0$ such that $|f(x)-L|< \epsilon$ whenever $0<x-a<\delta$.
\end{definition}

\indent
As an important remark, the theorems discussed in the previous article still holds true for one-sided
limits where "$x-a$" is replaced by "$x-a^{+}$" or "$x-a^{-}$".
to:
if for every {$\epsilon >0$}, no matter how small, there exists a {$\delta>0$} such that {$|f(x)-L|< \epsilon$} whenever {$0<x-a<\delta$}.

%lframe
text-align=center%As an important remark, the theorems discussed in the [[ LimitOfAFunction | previous article ]] still holds true for one-sided limits where "{$x-a$}" is replaced by "{$x-a^{+}$}" or "{$x-a^{-}$}".

\begin{example}
Evaluate
the following limits:
October 26, 2010 by matthew_suan -
Changed lines 3-13 from:
%rframe width=250px border='0px solid black' padding=2px%Attach:one_sided_limits.jpg | '''Figure 1: Plot of the function {$f(x)=\sqrt{1+x}$}.'''

On the other hand, one sided limits are also considered here. This kind of limits are important because when there is a discontinuity of the function on either side of $a$, then the two-sided limits is not true at all. It may be true on one side but it can't be on the other side. As an illustration, consider the function {$f(x)=\sqrt{1+x}$} which is plotted in Figure 1.\\

Note that the value of the function is imaginary at points {$x<-1$} as seen in the figure. 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 {$a$}, thus, the two sided limit does not make sense in this type of configuration. However, if we consider the values of {$x$} that are greater than -1, then we find that the values of {$f$} are defined! See, so this means that the one-sided limit {$\lim_{x \to -1^{+}}\sqrt{1+x}$} can be considered.\\

Obviously,
there are two types of one-sided limits. The first type is when the value of {$x$} approaches from the right of {$a$}. This is so because we are considering the values of {$x$} to the right of {$a$}. This only happens when {$f$} is properly defined on that side. In limit notation, this goes like,

(:table border=3 cellpadding=3 cellspacing=0 align=center:)
to:
!!Why is there a single-sided limits?

%rframe width=200px border='0px solid black' padding=2px%Attach:one_sided_limits.jpg | '''Figure 1: Plot of the function {$f(x)$}.'''

One sided limits are considered and is important because when there is a discontinuity of the function on either side of {$a$}, then the two-sided limits is not true at all. It may be true on one side but it can't be on the other side. As an illustration, consider the function {$f(x)=\sqrt{1+x}$} which is plotted in Figure 1.\\

Note that the value of the function is imaginary at points {$x<-1$} as seen in the figure. 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 {$a$}, thus, the two sided limit does not makes sense in this type of configuration. However, if we consider the values of {$x$} that are greater than -1, then 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 two-sided limits but also one-sided limits as well which comes in two types. \\
[[<<]]
!!Classifications
of One-Sided Limits

(:table
(:cellnr
bgcolor=#d4d7ba colspan=14 align=center:) '+'''Two Types of One-Sided Limits'''+'
Changed lines 18-19 from:
'+{$$\lim_{x \to a^{+}} f(x)=L$$}+'
to:
*[[#t1 | Right-Handed Limit]]
*[[#t2
| Left-Handed Limit]]
Changed lines 22-24 from:

which
is read as "L is the limit of {$f(x)$} as {$x$} approaches a from the right". Likewise, when "L is the limit of {$f(x)$} as {$x$} approaches a from the left, then we may write it as
to:
As mentioned, there are two types of one-sided limits. The first type is when the value of {$x$} approaches from the right of {$a$}. This is so because we are considering the values of {$x$} to the right of {$a$}. This only happens when {$f$} is properly defined on that side. In limit notation, this goes like,

Changed line 28 from:
'+{$$\lim_{x \to a^{-}} f(x)=L$$}+'
to:
'+{$$\lim_{x \to a^{+}} f(x)=L$$}+'
which is read as "L is the limit of {$f(x)$} as {$x$} approaches a from the right". Likewise, when "L is the limit of {$f(x)$} as {$x$} approaches a from the left, then we may write it as

(:table border=3 cellpadding=3 cellspacing=0 align=center:)
(:cellnr:)
'+{$$\lim_{x \to a^{-}} f(x)=L$$}+'
(:tableend:)
Changed lines 42-43 from:
\begin{definition}[Right-Hand Side Limit]
Let $f$ be a function defined at every point in some open interval (a,b). Then \textbf{the limit of f(x) as x approaches a from the right is L}, notated as
to:
[[#t1]]
!!Right-Hand
Side 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 {$L$}, notated as
October 26, 2010 by matthew_suan -
In the previous lessons, we are concerned with the values of {$x \neq a$} when considering the limit of {$f$} as it approaches {$a$} in an open interval {$I$} that are close and yet, contains {$a$}. We did work out on those limits without being particular to which direction the approach is coming. We just consider both values of {$x$} coming from the left and right of {$a$}. This kind of mechanism is attributed to two-sided limits. \\

%rframe width=250px border='0px solid black' padding=2px%Attach:one_sided_limits.jpg | '''Figure 1: Plot of the function {$f(x)=\sqrt{1+x}$}.'''

On the other hand, one sided limits are also considered here. This kind of limits are important because when there is a discontinuity of the function on either side of $a$, then the two-sided limits is not true at all. It may be true on one side but it can't be on the other side. As an illustration, consider the function {$f(x)=\sqrt{1+x}$} which is plotted in Figure 1.\\

Note that the value of the function is imaginary at points {$x<-1$} as seen in the figure. 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 {$a$}, thus, the two sided limit does not make sense in this type of configuration. However, if we consider the values of {$x$} that are greater than -1, then we find that the values of {$f$} are defined! See, so this means that the one-sided limit {$\lim_{x \to -1^{+}}\sqrt{1+x}$} can be considered.\\

Obviously, there are two types of one-sided limits. The first type is when the value of {$x$} approaches from the right of {$a$}. This is so because we are considering the values of {$x$} to the right of {$a$}. This only happens when {$f$} is properly defined on that side. In limit notation, this goes like,

(:table border=3 cellpadding=3 cellspacing=0 align=center:)
(:cellnr:)
'+{$$\lim_{x \to a^{+}} f(x)=L$$}+'
(:tableend:)

which is read as "L is the limit of {$f(x)$} as {$x$} approaches a from the right". Likewise, when "L is the limit of {$f(x)$} as {$x$} approaches a from the left, then we may write it as

(:table border=3 cellpadding=3 cellspacing=0 align=center:)
(:cellnr:)
'+{$$\lim_{x \to a^{-}} f(x)=L$$}+'
(:tableend:)

The symbol "{$x \to a^{+}$}" means that {$x$} approaches {$a$} through values greater than {$a$} while the symbol "{$x \to a^{-}$}" means that {$x$} approaches {$a$} through values less than {$a$}. With all these in mind, let us now define those one-sided limits formaly.

\begin{definition}[Right-Hand Side Limit]
Let $f$ be a function defined at every point in some open interval (a,b). Then \textbf{the limit of f(x) as x approaches a from the right is L}, notated as

\begin{equation}
\lim_{x \to a^{+}} f(x)=L
\end{equation}
if for every $\epsilon >0$, no matter how small, there exists a $\delta>0$ such that $|f(x)-L|< \epsilon$ whenever $0<x-a<\delta$.
\end{definition}

\begin{definition}[Left-Hand Side Limit]
Let $f$ be a function defined at every point in some open interval (c,a). Then \textbf{the limit of f(x) as x approaches a from the left is L}, notated as

\begin{equation}
\lim_{x \to a^{-}} f(x)=L
\end{equation}
if for every $\epsilon >0$, no matter how small, there exists a $\delta>0$ such that $|f(x)-L|< \epsilon$ whenever $0<x-a<\delta$.
\end{definition}

\indent
As an important remark, the theorems discussed in the previous article still holds true for one-sided limits where "$x-a$" is replaced by "$x-a^{+}$" or "$x-a^{-}$".

\begin{example}
Evaluate the following limits:

\begin{equation}
\lim_{x \to 1^{+}} \frac{x-1}{\sqrt{x^{2}-1}}
\end{equation}

\end{example}

\indent
Notice that if we follow directly theorems discussed before by substituting directly 1 to the function, it gives zero at the denominator. To avoid these, let us rationalize the denominator. Note that rationalizing denominator is not a standard option in solving one-sided limits. The use of such is on a case to case basis. Well anyway, let's proceed.
\newline{}
\indent
First, let us rationalize the function

\begin{eqnarray}
f(x) & = & \frac{x-1}{\sqrt{x^{2}-1}} \\
& = & \frac{x-1}{\sqrt{(x+1)(x-1)}}\cdot \frac{\sqrt{(x-1)(x+1)}}{\sqrt{(x+1)(x-1)}} \\
& = & \frac{(x-1)\sqrt{x^{2}-1}}{\sqrt{(x+1)^{2}(x-1)^{2}}} \\
& = & \frac{(x-1)\sqrt{x^{2}-1}}{(x+1)(x-1)} \\
& = & \frac{\sqrt{x^{2}-1}}{(x+1)}
\end{eqnarray}

\noindent
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.

\begin{eqnarray}
\lim_{x \to 1^{+}} f(x) & = & \lim_{x \to 1^{+}} \frac{\sqrt{x^{2}-1}}{(x+1)} \\
& = & \frac{\sqrt{1^{2}-1}}{1+1} \\
& = & \frac{0}{2} \\
& = & 0
\end{eqnarray}

Thus, we have successfully solved for the limit of $f(x)$. In a broader sense, the solutions and techniques employed in solving two-sided limits are still applicable for one-sided limits, as well as the theorems involved.

\end{document}