I have a piece-wise function similar to this:
f(x) = (-ln x)/3 (x>0)
The limit of this is infinity, it exists as the one sided limits exist for f(x) and both equal infinity (although can it be argued that lim x->0 -2/x DNE?).
So how does one classify this discontinuity, is it a removable or essential discontinuity? Why?
Any help is appreciated.
If lim(x->a) f(x) DNE, essential discontinuity at a.
If lim(x->a) f(x) = L =/= f(a) then it's removable.
Turns out, underneath the limit laws, my book also defines: lim (x->a) f(x)/g(x) = L/M if M=/=0, DNE if M=0, L=/=0.
Using this definition, lim (x->0) -2/x DNE, therefore two sided limits are not equal, therefore the discontinuity is essential. Is this correct?