tailieunhanh - Handbook of mathematics for engineers and scienteists part 40

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 40', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Basic Concepts of Mathematical Analysis 241 3. If there exist lim f x and lim g x then x a x a lim f x g x lim f x lim g x x aL J x a x a lim cf x c lim f x c const x a x a lim f x g x lim f x lim g x x a x a x a f x lim f x lim f4 7a if g x 0 lim g x 0 . x a g x lim g x x a 7 x a 4. Let f x g x in a neighborhood of a point a x a . Then lim f x lim g x provided that these limits exist. 5. If f x g x h x in a neighborhood of a point a and lim f x lim h x b x a x a then lim g x b. x a These properties hold also for one-sided limits. . Limits of some functions. First noteworthy limit Second noteworthy limit Some other frequently used limits lim 1. x 0 x M 1 y e lim 1 x n 1 _ n x 0 x 1 cos x lim---5--_ x 0 x2 ta ex _ 1 x 0 x Lm intx _ 1 x 0 x anxn an 1xn 1 a1x a0 lim ---------r--- T x o bnxn - -b l _ 1 x 0 x lim 1 _ln a x 0 x Umianhi _ 1 x 0 x bn ixn 1 bix bo j. arcsin x i x 0 X ln 1 x lim---------- 1 x 0 x j. arcsinh x i x 0 x _ an _ bn arctan x lim x 0 x loga 1 x _ . ----- _ loga e x 0 x J. arctanh x 1 x 40 x 1 2 lim xa ln x _ 0 lim x a ln x _ 0 lim xaex _ 0 lim xx _ 1 x 0 x o x o x 0 where a 0 and bn 0. See Paragraph where L Hospital rules for calculating limits with the help of derivatives are given. . Asymptotes of the graph of a function. An asymptote of the graph of a function y f x is a straight line whose distance from the point x y on the graph of y f x tends to zero if at least one of the coordinates x y tends to zero. 242 Limits and Derivatives The line x a is a vertical asymptote of the graph of the function y f x if at least one of the one-sided limits of f x as x - a 0 is equal to to or -to. The line y kx b is an oblique asymptote of the graph of y f x if at least one of the limit relations holds lim f x - kx - b 0 or lim f x - kx - b 0. If there exist finite limits lim f x k lim f x - kx b x œ x x œ then the line y kx b is an oblique asymptote of the graph for x to in a similar way one defines an asymptote for x -to .