A question about using the squeeze theorem? If you have the inequalities not equal to, but the function is just strictly between two other functions, can you use the squeeze theorem?
Remember,  if you have a fact "A", then you also have the fact "A or B",  for any fact B.   So, in this case,  if you have "$x&lt;y$" then you have $x\le y$ because that's just shorthand for $x&lt;y$ OR $x=y$,  and since we have $x&lt;y$, then we have $x\le y$.  So yes,  strict inequality is just fine, as that's even better than less than or equal to.
Sorry for the late answer but just answering in case anyone else has the same doubt.
by looking at the question ,it seems that the op is wondering if $f(x) &lt; g(x) &lt; h(x)$ how can it be possible that the limits are equal as the functions are not allowed to be equal
The fact is $$f(x) &lt; g(x) \implies \lim_{x\to a}f(x) \leq \lim_{x\to a}g(x) $$
this can be easily proved by taking h(x)=f(x)-g(x) and h(x)&lt;0,now apply $\epsilon-\delta$ definition ,u cant rule out the possibility of $lim_{x\to a} h(x)=0$ and now u can use limit law to split f(x) and g(x) and hence proving the original statement
Several examples showing such scenario are provided in comments section.
Now coming to squeeze theorem,now u are possibly understanding that even if u have strict inequalities for the functions,it translates to having non strict inequalities for the limit and hence when the lower and upper bound for $\lim_{x \to a} g(x)$ are equal and it is allowed to take that value,it must take that value
alternatively, if u want to prove squeeze theorem for strict inequality from scratch ie. from $\epsilon -\delta $  definition , then proceed as shown in https://en.wikipedia.org/wiki/Squeeze_theorem#Proof and just remove equality wherever necessary