x {\displaystyle \mathbb {R} \cup \{-\infty ,+\infty \}} ⁡ 2 and , = R is a neighborhood of x ≤ 0 ± {\displaystyle f} 0 0 − ∞ {\displaystyle x_{0}} {\displaystyle a+\infty } ) ∞ ( {\displaystyle {\overline {\mathbb {R} }}} is a limit of 1 He concludes that the responses are consistent with a link between historical development and individual development, although the cultural context is different.[7]. R f {\displaystyle x=+\infty } ∞ {\displaystyle \mathbb {R} } approaches. {\displaystyle \mathbb {R} } 1 ⁡ ∞ However, in contexts where only non-negative values are considered, it is often convenient to define This page was last edited on 9 November 2020, at 06:22. The notion of the neighborhood of is often defined as {\displaystyle a_{n}} {\displaystyle -\infty } R f − … 1 0 ∞ {\displaystyle x} a {\displaystyle 1/x} a + x {\displaystyle {\overline {\mathbb {R} }}} {\displaystyle +\infty } for all {\displaystyle f(x)=x^{2}\sin(1/x)} that agrees with the usual length of intervals, this measure must be larger than any finite real number. / {\displaystyle -\infty } + {\displaystyle 1/0} ∞ {\displaystyle 1/0=+\infty } ¯ ∈ {\displaystyle 1/f} x (called indeterminate forms) are usually left undefined. exp ). (for the latter function, neither , then it need not be the case that The graph of this function has a horizontal asymptote at y = 0. ) N ( , n Geometrically, when moving increasingly farther to the right along the is often written simply as This article is about the extension of the reals by, Learn how and when to remove this template message, "The Definitive Glossary of Higher Mathematical Jargon — Infinite", "Section 6: The Extended Real Number System", https://en.wikipedia.org/w/index.php?title=Extended_real_number_line&oldid=981472189, Articles lacking in-text citations from May 2014, Creative Commons Attribution-ShareAlike License, Computer representations of extended real numbers, see, This page was last edited on 2 October 2020, at 14:34. : {\displaystyle [-\infty ,+\infty ]} ∞ can be continuously extended to {\displaystyle -\infty } / x for x You can easily convince yourself of this by tapping into your calculator the partial sums and so on. {\displaystyle [0,1]} For example, for you get and for you get This is why mathematicians say that the sum divergesto infinity. − as x + − ( Joel Lehmann describes the process of distinguishing between different sum concepts as building a bridge over a conceptual crevasse: the confusion over divergence that dogged 18th-century mathematics. {\displaystyle \{a_{n}\}} ±