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Mathematics for Physics

326 Limit

A value that a function approaches as its variable approaches a point that is used to define continuity of a mapping at a point.

definition [d] (Limit) A real number \(L\) such that the values of \(f(x)\) can be made arbitrarily close to \(L\) by taking \(x\) sufficiently close to \(a\) on either side, but not equal to \(a\), written \(\lim_{x \to a} f(x) = L\).

where

  • \(f\) is a real-valued function.
  • \(a\) is the point approached.
  • \(L\) is the limit value.
  • \(x\) is the independent variable.

Note:

  • \(f\) need not be defined at \(x = a\).

definition [d] (Limit) A real number \(L\) associated with a function \(f\) defined on some open interval containing \(a\), except possibly at \(a\) itself, such that for every \(\epsilon > 0\) there exists \(\delta > 0\) with

  • \(0 < \lvert x - a \rvert < \delta\) implies \(\lvert f(x) - L \rvert < \epsilon\) ,

written \(\lim_{x \to a} f(x) = L\).

where

  • \(f\) is a real-valued function.
  • \(a\) is the point approached.
  • \(L\) is the limit value.
  • \(\epsilon > 0\) is an arbitrary tolerance on \(\lvert f(x) - L \rvert\).
  • \(\delta > 0\) is a corresponding restriction on \(\lvert x - a \rvert\).

Note:

  • the condition excludes the point \(x = a\).