Question

Give a piecewise function that changes behaviour at x=3, x=1, and x=4 such that at each of these three points, the left and right hand limits exist, but such that the limit exists at exactly one of these points. Can someone suggest how to start this?

Answer 1

*** limit exists at exactly one of these points. ***
this means the left-hand limit equals the right-hand limit

you could have a point where both limits exist, but *are different*

example: f(x) =1 for x<1
= 2 for x>1
\[\lim_{x \rightarrow 1^{-}} f(x) = 1\]
\[\lim_{x \rightarrow 1^{+}} f(x) = 2\]