section_path: Types of limits/Nonstandard analysis
section_name: Nonstandard analysis
paragraph_text: Nonstandard analysis
  
  In non-standard analysis (which involves a hyperreal enlargement of the number system), the limit of a sequence (a n) {\displaystyle (a_{n})} can be expressed as the standard part of the value a H {\displaystyle a_{H}} of the natural extension of the sequence at an infinite hypernatural index n=H. Thus,
  
  Here, the standard part function "st" rounds off each finite hyperreal number to the nearest real number (the difference between them is infinitesimal). This formalizes the natural intuition that for "very large" values of the index, the terms in the sequence are "very close" to the limit value of the sequence. Conversely, the standard part of a hyperreal a = {\displaystyle a=} represented in the ultrapower construction by a Cauchy sequence (a n) {\displaystyle (a_{n})}, is simply the limit of that sequence:
  
  In this sense, taking the limit and taking the standard part are equivalent procedures.