Question

Why the squeezing principle is only valid in an open interval for the domain of the studied functions?

Answer 1

I am not sure I understand your question. but in using the "squeezing" (aka "pinching") rule to solve limits, it is only necessary to consider the vicinity that the limit variable is approaching. In trig functions that will often be in the first and fourth quadrants around theta = 0.

After all, we aren't really interested in the squeezing functions elsewhere.
They don't need to confine the function we are interested in elsewhere, and often they don't.

Answer 2

Thank you Lars. Maybe I didn't explain in the best way. My question was more way we use <= and >= (g(x)<=f(x)<=h(x)) and not < >. (g(x)<f(x)<h(x)).

But now I see that it would squeeze if we didn't have the = too.

Thank you.