(Squeeze Theorem)
Let f,g, and h satisfy f(x) <= g(x) <= h(x) for all x in some common domain A. If lim as x-->c of f(x) = L and lim as x-->c of h(x) = L at some limit point c of A, show lim as x-->c g(x) = L as well.
(Squeeze Theorem)
Let f,g, and h satisfy f(x) <= g(x) <= h(x) for all x in some common domain A. If lim as x-->c of f(x) = L and lim as x-->c of h(x) = L at some limit point c of A, show lim as x-->c g(x) = L as well.
Given $\displaystyle \epsilon>0,$ since $\displaystyle f\to\ell$ and $\displaystyle h\to\ell$ as $\displaystyle x\to c,$ it follows that $\displaystyle \ell-\epsilon<f<\ell+\epsilon$ and $\displaystyle \ell-\epsilon<h<\ell+\epsilon.$ Besides, as $\displaystyle x$ closes to $\displaystyle c,$ it's $\displaystyle f\le g\le h,$ then $\displaystyle \ell-\epsilon<f\le g\le h<\ell+\epsilon,$ hence $\displaystyle |g-\ell|<\epsilon,$ therefore $\displaystyle g\to\ell$ as $\displaystyle x\to c.\quad\blacksquare$