Question

How do you classify the discontinuities (if any) for the function `f(x) =(x-4)/(x^2-16)`?

Answer 1

Can be rewritten as `f(x)=((x-4))/((x+4)(x-4))`

Both `x=-4andx=4` are 'forbidden' values for `x`

If `x!=4` we may cancel `=1/(x+4)`

The discontinuity at `x=4` is a removable discontinuity, because

`lim_(xrarr4) f(x)` exists (it is `1/8`)

The discontinuity at `x=-4` is non-removable.
(The limit as `xrarr-4` does not exist.)
More specifically, since the one sided limit as `x rarr -4^+` is infinite, this is an infinite discontinuity.

Note
When we say "the limit is infinite" we are really saying "the limit does not exist, because as `x rarr -4^+`, the function increases without bound".