By I.A. Maron

Desk of content material creation to mathematical research differentiation of services program of differential calculus to research of services indefinite integrals. simple tools of integration easy sessions of integrable capabilities the yes imperative purposes of the convinced critical fallacious integrals

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Taking advantage of the theorem on passing to the limit in inequalities, prove that Jim Xn=l if Xn=2n(Vn 2 +l-n). 13. Prove that the sequence Xi= Va; x3 = X2 o:c (-a-+-:v::--;r=a=+=V=a; ... =+=V=a '--·~~--,-·~~---- n radicals (a> 0) has the limit b=(V4a+l+l)/2. 14. Prove that the sequence with the general term I I I Xn = 3 + I + 32 + 2 + · · · + 3n + n has a finite 1imi t. 15. Prove that a sequence of lengths of perimeters of regular 2n-gorn inscribed in a circle tends to a limit (called the length of circumference).

A ) 1Im 2 2 2 X-+0 2 12 X-+0 X X-+0 X X X-+ . tanx-sinx 1. sinx(l-cosx) (b) IIm = Im -----=---'x ... o x3 X-+0 COSX•x3 I sinx 1-cosx . =II m - - · - - · - - x ... 0 cos x x x2 (c) Let us put l-x=z. Then Hence, X= 1-z and Z--+0 as X--+ I. :!. z 2 2 - =1·I I T I 2 -=:re II. m -2- = 1·Im . X-+I 1-X Z-+0 z l-+0 z 2 Note. 12. 5. ~~ (i ~xY: () I . e ln(l+x). Im 3x_1 ' x ... < ... 0 x lnx-1 (I")I"I m --. Je-+B x-e (b) lim(l+x)''<3x>; x _.. 0 (d) lim(l+k/xrx; • e4X-I (f} hm -1 - ; x .. 0 an x (h) Jim ex -:-e-:x ; x ..

2n-1 hm 2n +1=1. n-+rn (b) Let us find the absolute value of the difference lxn-3/51: 3n2+ I 31 8 15n2-I -5 = 5 (5n2- I)· Let 8 >0 be given. Choose n so that the inequality 8 5 (5n2 - I) <8' is fulfilled. Solving this inequality, we find n 2 8 +I. > 25e 5' n > ..!... /8+5e 5 V e • Putting N=E (; y-s~5g)' we conclude that at n >N lxn-3/51<8, which completes the proof. 5. Given a sequence with known that lim Xn = 1/3. 01). the general term Xn=;~+!. It is the number of points Xn lying out- -+ "' side the open interval L= (-}-1 o~o; ~ + 1~oo) · 44 Ch.

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Problems in Calculus of One Variable by I.A. Maron


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