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The limit does not exist.
sin(x)-cos(x) = (1)sin(x)+(-1)cos(x) so the range is sqrt((1)^2+(-1)^2)=1 and the domain is R <><><><><> The domain of sin x - cos x is [-infinity, +infinity]. The range of sin x - cos x is [-1.414, +1.414].
If I understand your question correctly, such a sequence is an = x cos(πx). It has neither an upper nor lower bound. It's divergent, but its limit is neither infinity nor negative infinity.
tan x = sin x / cos x sin x = x - (x^3/3!) + (x^5/5!) - (x^7/7!) + (x^9/9!) ... to infinity cos x = 1 - (x^2/2!) + (x^4/4!) - (x^6/6!) + (x^8/8!) .. to infinity tan 165 degrees = -0.267949192
Cos times Cos