Finding One-Sided Limits: The Simplest Way

[mooneh]

heey, i know how to find limits but i can't find limits from the left and from the right
can someone pleasezzzz show me the simplist way to do it
thx

 

[sadhu]

suppose
you want to find limit when x tends to a
substitute x with a+h
now find the directive limit for h tends to 0
you see that only difference in both limits is the sign of h,value remain same
so take underconsideration the sign and substitute 0 in function if it is defined for both sides

 

[mooneh]

can u give me an example ?

 

[HallsofIvy]

If you can find "limits", then "one-sided limits" should be easy!

Here's one easy example:
\lim_{x\rightarrow 1^+} x^2
Since x^2 has a (regular) limit, 1, the two one-sided limits (from the left and right) must be the same:
\lim_{x\rightarrow 1^+} x^2= \lim_{x\rightarrow 1^-} x^2= 1

Here's a slightly harder example:
\lim_{x\rightarrow 1^+} f(x)
where f(x)= x^2 if x< 1 and if f(x)= x+ 4 if x> 1.
Of course, \lim_{x\rightarrow 1^+} f(x) depends only on the value of the function for x> 1, this is exactly the same as
\lim_{x\rightarrow 1} x+ 4[/itex]<br /> which is 5.<br /> \lim_{x\rightarrow 1^+} f(x)= 5 <br /> Similarly<br /> \lim_{x\rightarrow 1^-} f(x)= \lim_{x\rightarrow 1} x^2= 1 <br /> In this case, since the two &quot;one-sided&quot; limits are different, the &quot;limit&quot; itself does not exist. Typically, you find one-sided limits <b>in order</b> to determine whether the &quot;limit&quot; itself exist and, if so, find the value of the limit. Also, typically, you find the one-sided limit by determining the &quot;limit&quot; for the function giving the value on <b>that</b> side of the point at which you are taking the limit.

 

[sadhu]

lim\sqrt{1-x}
x\rightarrow 1find right hand limit ,it is undefined because , if you by making x=1+h
then you see that root of negative no does not exist but left hand limit does exist and is 0