One sided limits

  • Thread starter mooneh
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  • #1
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heey, i know how to find limits but i cant find limits from the left and from the right
can someone plzzzzz show me the simplist way to do it
thx
 

Answers and Replies

  • #2
158
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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
 
  • #3
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can u give me an example ?
 
  • #4
HallsofIvy
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If you can find "limits", then "one-sided limits" should be easy!

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

Here's a slightly harder example:
[tex]\lim_{x\rightarrow 1^+} f(x)[/tex]
where f(x)= [itex]x^2[/itex] if x< 1 and if f(x)= x+ 4 if x> 1.
Of course, [itex]\lim_{x\rightarrow 1^+} f(x)[/itex] depends only on the value of the function for x> 1, this is exactly the same as
[tex]\lim_{x\rightarrow 1} x+ 4[/itex]
which is 5.
[tex]\lim_{x\rightarrow 1^+} f(x)= 5[/tex]
Similarly
[tex]\lim_{x\rightarrow 1^-} f(x)= \lim_{x\rightarrow 1} x^2= 1[/tex]
In this case, since the two "one-sided" limits are different, the "limit" itself does not exist. Typically, you find one-sided limits in order to determine whether the "limit" itself exist and, if so, find the value of the limit. Also, typically, you find the one-sided limit by determining the "limit" for the function giving the value on that side of the point at which you are taking the limit.
 
  • #5
158
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[tex]lim[/tex][tex]\sqrt{1-x}[/tex]
[tex]x\rightarrow 1[/tex]


find 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
 
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