squeeze theorem

Suppose you cannot find someone in a crowd, but you know they are walking arm in arm between two friends, and those two friends are both converging on the same exit. Then the person in the middle has no choice — they must be heading for that exit too. The squeeze theorem turns this intuition into a tool: if a function is trapped between two others that both head to the same limit, the trapped function must go there as well.

Formally: if g(x) <= f(x) <= h(x) for all x near a (you do not need it to hold at a itself), and if lim x->a g(x) = lim x->a h(x) = L, then lim x->a f(x) = L too. The two bounds must squeeze toward the SAME value L; if the lower and upper bounds head to different numbers, the theorem says nothing. It is especially valuable for functions that wiggle too wildly to handle directly.

The classic payoff is lim x->0 sin(x)/x = 1, a fact you cannot get by simply plugging in (that gives 0/0), but which falls out of squeezing sin(x)/x between cos(x) and 1. Another favorite is lim x->0 x^2 sin(1/x) = 0: the sine factor oscillates forever, yet it is pinned between -x^2 and x^2, both of which go to 0, so the product is forced to 0. The theorem lets rigorous bounds win where direct evaluation cannot reach.