Give example two function f and g are discontinuous but f/g is continuous .

Question

Sketch the graph of f/g .

Answer 1

Let f(x)=1/x, which is discontinuous at x=0, and g(x)=1/(x^2+x), which is discontinuous at x=0 and x=-1. (f/g)(x)=(x^2+x)/x=x+1, which is continuous for all x.

The red curve is f(x), the blue curve is g(x) and the green line is (f/g)(x).

There are asymptotes at x=0 (y-axis) and x=-1 (blue vertical line) for f and g, marking their discontinuities.