CalcNotes0203-page29 - (Section 2.3 Limits and Infinity I...

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Unformatted text preview: (Section 2.3: Limits and Infinity I) 2.3.29 FOOTNOTES 1. Infinity. • Infinity is not a number in the usual real number system that we will study in calculus. • The affinely extended real number system, denoted by ¡ or ¡ ¢ , ¢ £ ¤ ¥ ¦ , includes two points of infinity, one referred to as ¡ (or + ¡ ), and the other referred to as ¡ ¢ . (We are “adjoining” them to the real number system.) We obtain the two-point compactification of the real numbers. We never refer to ¡ and ¡ ¢ as real numbers, though. • Sometimes, ¡ and ¡ ¢ are treated as the same (we collapse them together and identify them with one another as ¡ ), and we then obtain the one-point compactification of the real numbers, also known as the real projective line. Then, we can write 1 = ¡ , and we can say that the slope of a vertical line is ¡ . • A point at infinity is sometimes added to the complex plane, and it typically corresponds to the “north pole” of a Riemann sphere that the complex plane can be thought of as wrapping...
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