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Find the lim(x- 2-),(x- 2+),(x- 2) given f(x) {(x-1)^2 if x2}

Customer Question

In words: Find the limit as x approched 2-, 2+ as f(x) is defined (x-1)^2 if x<2
2    ;   if x=2
-x+3  &n bsp;if x>2
Submitted: 11 years ago.
Category: Homework
Expert:  xarqi replied 11 years ago.
Are you looking for the one-sided limits as x approaches 2 from
below or above?
Is f(x) undefined except where x = 2?
Expert:  xarqi replied 11 years ago.
OK - you've answered the second point, thanks.

(I'm too eager I guess!)
Expert:  xarqi replied 11 years ago.
Assuming I interpreted the "2+" and "2-" correctly:
lim(x-> downward to 2) = -2 + 3 = 1
lim(x -> upward to 2) = (2-1)^2 = 1

As it happens, the function tends to 1 at x=2 from both sides,
however, it is discontinuous at x=2.
Customer: replied 11 years ago.
Reply to xarqi's Post: Can I ask you this question again better? I am looking for lim f(x) as x approaches 2- 2+ and 2 f(x) is defined
f(x)={(x-1)^2 if x<2
2 if x=2
-x+3 if x>2}
Expert:  xarqi replied 11 years ago.
It's just a terminology thing that is confusing me. I'm taking
"2-" to mean "tends to 2 from below", that is, it is a left-hand
limit; and conversely for "2+"

The question: lim(x->2), where no direction is given for the
approach to the discontinuity requires me to do a bit of reading
before offering a solution. On the one hand, since the limit from
either side is1, it might be reasonable to infer that the limit at 2
is also 1, but that is at odds with the fact that f(2) = 2.

I'll get back to you on that with luck.
Expert:  xarqi replied 11 years ago.
This seems to be a matter of definition.

Where the lefthand limit of a function = the righthand limit of a
function at some value x, the limit at x is defined.

It seems that despite f(2) equalling 2 in this instance, lim(x->2)
= 1!

Go figure!

Perhaps a specialist here will have a definitive statement to
make.

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