How to use Squeez theorem on Lim(x approaches 0) (2x^2 cos 1/x)???

Question

Lim(x approaches 0) (2x^2 cos 1/x)

Answer 1

To do this, you first note that for all x not exactly zero, cos (1/x) is between -1 and 1. Therefore, for x not equal to zero, -2x^2 ≤ 2x^2 * cos(1/x) ≤ 2x^2. Therefore, since the limits as x goes to zero of both -2x^2 and 2x^2 are zero, we know by the squeeze theorem that the limit as x goes to zero of 2x^2 * cos(1/x) is zero as well.