Thank you in advance, if you're up for it, there are a couple more I'm stuck on. This one just really bothers me because I put a couple pages and about an hour of thinking into it
limit as x approaches zero of (1-cos^4(x ))/(sin2x)^2 = ??
I can get to
(1-cos^4(x ))/(sin2x)^2 = (1-cos^2x)(1+cos^2x) / ((2sinxcosx)(2sinxcosx))
the denominator might be wrong sin2x=2sinxcosx .
Or alternatively,
(1-cos^4(x ))/(sin2x)^2 = (1+cos^2(x ))(sin^2(x ))/sin^2 (2x)
Thanks for looking. I can't get it past there. The idea is to get to a point where things can be simplified by the following:
the lim as x approaches zero of sinx/x = 1
the lim as x approaches zero of sinx=0
the lim as x approaches zero of cosx=1
that's ridiculously more straightforward than I was expecting.
All the textbook question in this chapter for solving limits have been of the following format (no differentiating, just rearanging):
Find the limit as x approaches zero of (sin^2xcosx)/(1-cosx)
= lim as x approaches zero of (sin^2xcosx(1+cosx))/(1-cos^2x)
= lim as x approaches zero of (cosx(1+cosx))
=1(1+1)
=2
The rest of my limit questions make a lot more sense with a differential approach :P THanks again!