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x->0
2+cosx = 3 - (1/2)x^2 +o(x^2)
(2+cosx)/3 = 1 - (1/6)x^2 +o(x^2)
((2+cosx)/3)^x
=e^[x.ln((2+cosx)/3)]
=e^[x.ln(1 - (1/6)x^2 +o(x^2))]
=e^{ x. [- (1/6)x^2 +o(x^2)] }
=e^[-(1/6)x^3 +o(x^3)]
((2+cosx)/3)^x -1 =-(1/6)x^3 +o(x^3)
lim(x->0) (1/x^3) [ ((2+cosx)/3)^x - 1]
=lim(x->0) -(1/6)x^3/ x^3
-1/6
2+cosx = 3 - (1/2)x^2 +o(x^2)
(2+cosx)/3 = 1 - (1/6)x^2 +o(x^2)
((2+cosx)/3)^x
=e^[x.ln((2+cosx)/3)]
=e^[x.ln(1 - (1/6)x^2 +o(x^2))]
=e^{ x. [- (1/6)x^2 +o(x^2)] }
=e^[-(1/6)x^3 +o(x^3)]
((2+cosx)/3)^x -1 =-(1/6)x^3 +o(x^3)
lim(x->0) (1/x^3) [ ((2+cosx)/3)^x - 1]
=lim(x->0) -(1/6)x^3/ x^3
-1/6
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