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?
 A: 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 "$x<y$" then you have $x\le y$ because that's just shorthand for $x<y$ OR $x=y$,  and since we have $x<y$, then we have $x\le y$.  So yes,  strict inequality is just fine, as that's even better than less than or equal to.
A: As Alan points out, strict inequality is fine. I think there are some elementary, intuitive observations that needs to be made. Suppose you have functions $f$ and $g$ such that some limit $\lim f$ and $\lim g$ exists and coincide. Suppose moreover that $f < h < g$. Taking limits of all functions you get $$\lim f \le \lim h \le \lim g = \lim h$$ What can you say about $\lim h$?
A: 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 $f(x) < g(x) < h(x)$ how can it be possible that the limits are equal as the functions are not allowed to be equal
The fact is $$f(x) < g(x) \implies \lim_{x\to a}f(x) \leq \lim_{x\to a}g(x) $$
this can be easily proved by taking h(x)=f(x)-g(x) and h(x)<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
