Find the limit using the squeeze theorem

In this video we solve the limit x^2 + cos^3(1/x)/x^2 + sin(1/x) as x goes to 0 using the squeeze theorem!

[balinox]

you solve it by squeeze theorem
I solve it by what my hearts tells me
We are not the same

[rasmodii]

It's true that ∀x∈ℝ, x²+cos³(1/x)≤x²+1 and x²+sin(1/x)≥x²-1, but the quotient isnt always true (for -1<x<1 it's actually false), but though the limit remains correct (because the inequality is true for x≥1), doing this operation on limits without giving precision on why we're allowed to to it in an educational video isn't the best of ideas