We first need to determine lower/upper functions. We’ll start off by acknowledging that provided x≠0
(which we know it won’t be because we are looking at the limit as x→0)
we will have,
−1≤sinπx≤1
Now, simply multiply through this by x4 to get:
−x4≤sinπx≤x4
Before proceeding note that we can only do this because we know that x4> for x≠0. Recall that if we multiply through an inequality by a negative number we would have had to switch the signs. So, for instance, had we multiplied through by x3 we would have had issues because this is positive if x>0 and negative if x<0.
Now, let’s get back to the problem. We have a set of lower/upper functions and clearly,
limx→0x4=limx→0−x4=0
Therefore, by the Squeeze Theorem we must have,
limx→0x4sinπx=0.