One sided limits

1. Jan 1, 2008

mooneh

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

2. Jan 1, 2008

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. Jan 1, 2008

mooneh

can u give me an example ?

4. Jan 1, 2008

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

5. Jan 1, 2008

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

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

Last edited: Jan 1, 2008