I'm not sure if I fully understand the removable discontinuity concept. Can someone check this over for me please?

f(x)=

{2cos(x/2), x<0

{ln(x), for 0<x<1

{2x^3 - x - 1, for x>1

I know that the lim of 2cos(x/2) as x-->0- = -2

I know that the lim of 2x^3 - x - 1 as x-->0+ = -16pi^3 +2pi - 1

So is it correct to say that the lim of ln(x) as x approaches 0 from the right DNE because ln(x) is undefined when x = 0

Therefore f(x) is discontinuous at x = 0, because the function is undefined at x = 0 and because the limit does not exist.

Would it also be correct to say that this function is an example of a removable discontinuity, if we redefine f(x)

to be

{2cos(x/2), x<0

{-2, 0<x<1

{-2, x>1

How many changes are you allowed to make to a function f(x) in order to make it into a removable discontinuity?