by HallsofIvy » Wed Dec 11, 2019 9:07 am
Ouch! That's not easy!
The first thing I would try is just setting x= 0 in the given function. Unfortunately that makes both numerator and denominator 0 so the fraction is "undetermined" and we need to use another method to find the limit. The next thing I would try is to "rationalize the denominator". The denominator is [tex]\sqrt{1+ sin^2(2x)}- \sqrt{1+ sin^2(x)}[/tex] so multiply both numerator and denominator by [tex]\sqrt{1+ sin^2(2x)}+ \sqrt{1+ sin^2(x)}[/tex].
[tex]\frac{x tan(3x)\left(\sqrt{1+ sin^2(2x)}+ \sqrt{1+ sin^2(x)}\right)}{sin^2(2x)- sin^2(x)}[/tex].
Now use the trig identities [tex]sin(2x)= 2sin(x)cos(x)[/tex] so that [tex]sin^2(2x)- sin^2(x)= 4sin^2(x)cos^2(x)- sin^2(x)= sin^2(x)(4cos^2(x)- 1)[/tex] and [tex]sin^3(x)= \frac{3}{4}sin(x)- \frac{1}{4}sin(3x)[/tex] (I got that by looking up an identity for sin(3x))