Determine the limit
lim x->0 1-cos^2(3x)/x^2
lim x->0 1-cos^2(3x)/x^2
Since it is a situation of 0/0, we can use l'Hôpital's rule by differentiating both numerator and denominator to get:
lim x->0 1-cos^2(3x)/x^2
=lim x->0 6sin(3x)/2x
Since it is still a situation of 0/0, we can apply l'Hôpital's rule again:
=lim x->0 18cos(3x)/2
= 18*1/2
= 9
If you have learned about Taylor series or MacLauin series, you can expand the numerator into a series, and evaluate accordingly:
lim x->0 1-cos^2(3x)/x^2
=lim x->0 1-(1-(3x)^2/2!+(3x)^4/4!-...)^2/x^2
=lim x->0 1-(1-9x^2+27x^4-...)/x^2
=lim x->0 (9x^2-27x^4)/x^2
=lim x->0 (9-27x^2+...)
=9