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    Pinching Theorem for Functions

    Alternate names

    pinching theorem for functions | sandwich rule | sandwich theorem for functions | squeeze lemma | squeezing theorem for functions

    Theorem

    Let x_0 be a number in the open interval (a, b) and let f(x), g(x) and h(x) be real-valued functions that satisfy f(x)<=g(x)<=h(x) for all x in (a, b), except perhaps at x_0. Then if the limits lim_(x->x_0) f(x) and lim_(x->x_0) h(x) exist and are both equal to L, the limit lim_(x->x_0) g(x) is given by lim_(x->x_0) g(x) = L. The theorem also applies for limits to infinity. In particular, let f(x)<=g(x)<=h(x) for all x in (a, ∞). Then if lim_(x->∞) f(x) = L = lim_(x->∞) h(x), the limit lim_(x->∞) g(x) is given by lim_(x->∞) g(x) = L, and the analogous statement for -∞ also holds.

    Details

    closed interval | limit of a function

    squeeze theorem for sequences

    Associated people

    Archimedes | Eudoxus of Cnidus | Carl Friedrich Gauss

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