What are the removable and non-removable discontinuities, if any, of #f(x)=(x^2-100) / (x+10) #?

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Answer 1 · 2016-03-19T22:33:40

The only discontinuity is at #x=-10#. It is removable.

#f(x) = (x^2-100)/(x+10)# is a rational function.

Rational functions are continuous on their domains.

The domain of #f# is #(-oo,-10) uu (-10,oo)#.

So, #f# is continuous except at #x=-10#.

The discontinuity is removable if and only if #lim_(xrarr-10) f(x)# exists. (Note that, "infinite limits" are limits that do not exist.)

#lim_(xrarr-10) (x^2-100)/(x+10)# has initial form #0/0# which is indeterminate. Reduce the ratio and try again.

#lim_(xrarr-10) (x^2-100)/(x+10) = lim_(xrarr-10) ((x+10)(x-10))/(x+10) #

# = lim_(xrarr-10) (x-10) = -20#

The limit exists, so the discontinuity is removable.