Show that f(x) = x^2 and g(x) = 1000x^2 grow at the same rate.

Question

Show that f(x) = x² and g(x) = 1000x² grow at the same rate.

Answer 1

First step is to take the limit of the ratio f(x)/g(x) as x approaches +∞. (note that f(x) and g(x) have to be positive for large values of x).

The ratio turns out to be 1/1000 (the x² terms cancel). If this value is a finite non-zero number (in this case 1/1000), then they grow at the same rate.