What's the limit as x approaches 0 in (x - sin x) / (x - tan x)?
And the limit as x approaches 0 in (sin x - x) / x^3?
Please explain how you got the answer, thank you!
And the limit as x approaches 0 in (sin x - x) / x^3?
Please explain how you got the answer, thank you!
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Use L'Hopitals for each of these since both give you an answer of 0/0:
lim x->0 (x - sin x)/(x - tan x) = lim x->0 (1 - cos x)/(1 - sec² x) = lim x->0 cos²(x)(1 - cos(x))/(cos(x) + 1)(cos(x) - 1)
lim x->0 -cos²(x)/cos(x) + 1 = -1/2
lim x->0 (sin x - x)/x^3 = lim x->0 (cos x - 1)/(3x^2) = lim x->0 (-sin x)/(6x) = lim x->0 -cos(x)/6 = -1/6
lim x->0 (x - sin x)/(x - tan x) = lim x->0 (1 - cos x)/(1 - sec² x) = lim x->0 cos²(x)(1 - cos(x))/(cos(x) + 1)(cos(x) - 1)
lim x->0 -cos²(x)/cos(x) + 1 = -1/2
lim x->0 (sin x - x)/x^3 = lim x->0 (cos x - 1)/(3x^2) = lim x->0 (-sin x)/(6x) = lim x->0 -cos(x)/6 = -1/6