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
[tex]\lim_{x\rightarrow 1} x+ 4[/itex]
which is 5.
$$\lim_{x\rightarrow 1^+} f(x)= 5$$
Similarly
$$\lim_{x\rightarrow 1^-} f(x)= \lim_{x\rightarrow 1} x^2= 1$$
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.

 

[sadhu]

$$lim$$$$\sqrt{1-x}$$
$$x\rightarrow 1$$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