Question

Question #db818

Answer 1

If I am interpreting the question correctly, it is asking for examples of two functions, one of which has left and right hand limits at a point which are not equal, and the other in which only one of the left and right hand limits exists at some point.

1) Let `f(x) = x/|x|`. Then `f(x) = {(-1" if "x<0), (1 " if "x>0):}`, meaning `lim_(x->0^-) = -1` and `lim_(x->0^+) = 1`.

2) We can define a function piecewise.

Let `f(x) = {(0" if "x <= 0), (1/xsin(1/x)" if x > 0):}`

Then `lim_(x->0^-)f(x) = 0`, but `lim_(x->0^+)f(x)` does not exist, as it oscillates with increasing frequency and amplitude as `x->0^+`.