notes Functions asymptotic to each other

Thomas' Calculus: Early Transcendentals

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Unformatted text preview: Functions asymptotic {m} to each other We say that HE} NHL?) 35 E -*E in case lim L3) = 1. ._.. slim] This may be read as: ﬂs] is asymptotic to [or behava like] ﬁre] as .1 tends to :1. Here a may be a number or no. If we have several pairs of functions with fills] m gills], then products or quotients of the fife] behave like the corraponding product or quotient of the gifts]. Ifc is a non-zero constant, then ﬂit] m c as s —+ a is the same as lim1._.El ﬁe] = c. For example, let a be I]. Then we know that sins m s, and coss m 1. Hence tans m s, and sin[sg]tan[sa] m 3:333 5 5 m]! 41’ 41' so that sinfs3 :1 tantaall _ lim 1. 1—4] 41:5 This would have been a mas to prove using de l’Hopital’s rule. The notion of asymptotic functions simpliﬁa many topics in this course, some of which will be taken up later on. You may want to refer to this supplement repeatedly. Improper integrals Consider I: ﬁre] is. First identify all trouble spots a: points at which ﬂit} fails to be defined or continuous [these may be end points or interior points, and if c or d is too it must be included]. Then look to the left and to the right near the trouble spot a, and suppose that ﬂs] has a deﬁnite sign there. For example, suppose that ﬁx] is positive and continuous to the left of c. Find a simple function 9(3) such that ﬁre] m Ug[s] as s —+ a, where U is a constant. Then the two integrals ffimidm and fem: either both converge or both diverge [by comparison, since each is bounded by a constant times the other near a]. Here is a speciﬁc example: DJ 1 —d:r. f. v; + I. The two trouble spots are D and no. But ...
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