Determining Limits Using the Squeeze Theorem

Some functions can't be evaluated directly at a limit point because they misbehave, usually by oscillating forever, like sin(1/x) near x = 0. You can't plug in 0, and you can't simplify your way out. But if you can sandwich that troublesome function between two well-behaved functions, and both of those functions squeeze down to the same value, the function in the middle has no room to go anywhere else. It gets pinned.

Mechanically, you need g(x) ≤ f(x) ≤ h(x) for all x near the point in question (not necessarily at the point itself), and you need the limit of g(x) and the limit of h(x) as x approaches that point to be equal, say both equal to L. Since f(x) is always caught between them, as the gap between g and h closes down to zero, f(x) is forced to close in on L as well. The bounds do the work; f(x) just gets carried along.

This matters because it's the main tool for limits that resist algebraic tricks like factoring or rationalizing, especially anything involving sine or cosine of a reciprocal. It also sets up the most important limit in the whole unit: the limit of sin(x)/x as x approaches 0, which gets proven by squeezing it between cos(x) and 1 near zero. That limit becomes the foundation for differentiating sine and cosine later, so the squeeze theorem isn't just a clever trick here — it's load-bearing for what comes next.