Question

How do you graph and find the discontinuities of `f(x) = 1/x`?

Answer 1

The most obvious discontinuity is when `x=0` because that is not defined.

Explanation:

If `x->0` from positive, `f(x)` will grow, or in the "language":
`lim_(x->0^+) f(x)=oo`

Almost he same goes for `x->0` from negative:
`lim_(x->0^-) f(x)=-oo`

If `x` gets greater and greater (both positive and negative) `f(x)` will be smaller and smaller, without actually reaching `0`, or:
`lim_(x->oo) f(x)=lim_(x->-oo) f(x)=0`

`x=0 and y=0` are called the asymptotes.
graph{1/x [-10, 10, -5, 5]}