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here's what i got:lim x-> inf, (x + sinx)/(x + cos x)

(1 + (lim x ->inf, sinx/x))/(1 + (lim x->inf, cosx/x))

note the "1"s came from x/x from the original limit.

And lim x->inf, cosx/x becomes 0.

but I don't understand how lim x->inf, sinx/x becomes 0.

sinx/x = {1 - x^2/3! + x^4/5! - ... + ...} since x approaches infinity, that series would be 1 - inf + inf - inf, eventually it is still infinity.

so, we have (1 + inf)/(1 +0) = inf, but the text says it approaches 1, so it much be something wrong i did with the sinx/x thing.

Could someone please explain?