How does
lim (x --> 0) [tan^2(4x)] / (2x)^2
equal 4?
Then, are you familiar with the limit:
Limit (x -> 0) sin x / x = 1?
Now, tan^2(4x) / (2x)^2 = [sin(4x)/(4x)] * [sin(4x)/(4x)] * [4/cos^2(4x)]
Use the product rule. As x approaches zero, so does 4x, so the limit can be broken up as the product of three limits. The first two will evaluate to 1 and the last will evaluate to 4.