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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