Thank you so much HPL! Yeah yeah it makes sense now.. One more thing though, how do you know whether a point is a point of discontinuity (open circle) or an asymptote?
Another very good question indeed, and this definitely is something that will be hard to distinguish between.
When you are provided with an equation, you should become
suspicious of a point of discontinuity under two situations:
1. when a y-value of your curve approaches zero as x approaches the asymptote but the curve looks like as if it's going to intersect the asymptote
2. when you have f(x)g(x) but the domains are not the same (a typical example is xlnx, where domain of y = x is defined for all real x, but the domain for y = lnx is only defined for x>0)
A point of discontinuity is essentially when the function is undefined for both x and y-values of this particular point. For example, lets raise the y = xlnx again. x= 0 would be undefined for lnx because x>0, y = 0 is undefined because it is impossible to get a result of 0 from a log function. Hence the point (0,0) would be a point of discontinuity. When you become suspicious that there is a point of discontinuity, substitute in x and y-values of that point to confirm whether or not the point is discontinuous.
Finding a discontinuous point is not easy and it will take practise to develop an instinct to become suspicious of the existence of such points. In general, whenever you see lnx involved, you should keep an eye out on possible points of discontinuity.
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