#lim_(x->0) ((1-sqrt(cos6x))/(x^2)) = # ?

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Answer 1 · 2016-09-25T22:31:58

#9#

#lim_(x->0) ((1-sqrt(cos6x))/(x^2))#
# (1-sqrt(cos6x))/(x^2) = (1-cos6x)/(x^2(1+sqrt(cos6x))#

but

#cos6x=1-2sin^2 3x# so

# (1-cos6x)/(x^2(1+sqrt(cos6x))) = (2sin^2 3x)/(x^2(1+sqrt(cos6x))#

Finally

# (1-sqrt(cos6x))/(x^2) = 2(9(sin(3x)/(3x))^2)/(1+sqrt(cos6x))#

then

#lim_(x->0) ((1-sqrt(cos6x))/(x^2)) = lim_(x->0) 2(9(sin(3x)/(3x))^2)/(1+sqrt(cos6x)) = 9#