Thelimitofsequences1. Given limnun= aand limnvn= b.Provethat:limnu1vn + u2vn1 + + unv1n= abUsingtheaboveresulttoprove limnu1 + u2 + + unn=aandifun>0thenlimnnu1u2 un= a2. Findthelimitsofthefollowingsequences(a) an=1n+1+ +12n(b) an=n

k=11n2+2k(c) an=1n3n

k=1k[kx],withx Rand[]isintegerportion.(d) an=n

k=1arctan1k2+k+1(e) an=n

k=1arctan12k23. Considertheconvergenceofthefollowingsequences.(a) an= 1 +122+132+ +1n2(b) an= 1 +122+133+ +1nn(c) an=_1 12__1 14_ _1 12n_(d) an=_1 +12__1 +14_ _1 +12n_(e) an= 2n +11+12+ +1n(f) an= 2n + 1 +11+12+ +1n(g) an= 1 +12+13+ +1n 1 ln n(h) an= 1 +12+13+ +1n 1+1n ln n(i) an=n + 12n+1_2 +222+ +2nn_andfindthelimitofthissequence.4. Given {an}istherecurrentsequences.1(a) a1= 1, an+1=2(2an + 1)an + 3, (n N)(b) a1= 1, a2= 2, an+1= an1 +an, (n 2)(c) a1= 0, a2=12, an+1=13(1 + an + a3n1), (n 2)(d) a1= 2, an+1= 2 +13 +1an, n 1(e) a1=2, an+1=2 + an(f) a1= a, a2= b, an=an1 + an22(g) a1= a, a2= b, an= an1an2Toprove {an}isconvergentandfindthelimit.5. Given {an}is boundedsatisfies an+1 an 12n, n N. Provethat {an}isconvergent.6. Given {an}isboundedsatisfiesan+1 12(an + an1), n N.Provethat {an}isconvergent.7. Given {an}istherecurrentsequences.a1= b, an+1= a2n + (1 2a)an + a2Find a so that the sequence is convergent and operation of the limit in this case.8. Given {an}istherecurrentsequences.a0= 0, a1= 1, an= 2an1an2 + 1ProvethatA = 4anan2 + 1issquarenumberforalln 2.9. ProvetheoremStolz:Letsequencesan, bnsatisfies(i) bnisincreasingand limnbn= +.(ii) limnanan1bnbn1= l.Provethat limnanbn= l.UsingtheoremStolztofindthelimitofthefollowingsequences.(i) limn1n_1 +12+ +1n_(ii) limn1k+ 2k+ + nknk+1withk N(iii) limn1 +12+ +1nln n2(iv) limn_1nk(1k+ 2k+ + nk) nk + 1_withk N10. Findpartiallimit,inferiorlimitandsuperiorlimitofthefollowingsequences.(i) an= (1)n1_2 +3n_(ii) an= (1)n_1 +1n_n(iii) an= cosn_2n3_(iv) an=(1 + cos n) ln 3n + ln nln 2n(v) an=2n27_2n27_,[]isintegerportion.11. Given {un}ispositiveandboundedsatisfyingun+m unumforalln, m.Provethatunconverges.3Thelimitoffunctionandcontinuousfunction1. Thecomputationofthefollowinglimits.(a) limx031 + 3x 31 2xx + x2(b) limx_x2+ 1 x21_(c) limx051 + 3x41 2x31 + x 1 + x(d) limx0na + x na xx, n N(e) limx03a2+ ax + x23a2ax + x2a + x a x(f) limx0_1 + x2+ x_n_1 + x2x_nx(g) limx0na + x na xx,a > 0(h) limx0n1 + ax k1 + bxx,a > 02. Thecomputationofthefollowinglimitsbyusingthefundamentallimits.(a) limx0_1 + tan x1 + sin x_ 1sin x(b) limx+_sin1x+ cos1x_x(c) limx0_1 + x2x1 + x3x_1x2(d) limx0_cos xcos 2x_1x2(e) limx2(tan x)tan 2x(f) limx0_tan_4+ x__cot 2x(g) limx0_cos x_1x2(h) limx4_sin 2x_tan22x3. Letf(x) =_0 ifxisirrationalnumber,1 ifxisrationalnumber.Showthatlimxaf(x)doesnotexistforanya.4. Provethat limxcos x, limxsin x, limxtan xarentexistence.5. Considercontinuityofthefollowingfunctions.(a) f(x) =_0 ifx I,sin |x| ifx Q.(b) f(x) =_x21 ifx I,0 ifx Q.(c) f(x) =___| sin x|xifx = 0,a ifx = 0.(d) f(x) =_cos2_1x_ifx = 0,a ifx = 0.6. Survey continuity and classification of discontinuity points of the following func-tions.4(a) f(x) =x2x(b) f(x) = e1x(c) f(x) =_x ifx1ln x ifx > 1.(d) f(x) = |2x 3|2x 3(e) f(x) =_1xifx = 01 ifx = 0.7. Surveyuniformlycontinuityofthefollowingfunctionon(0, 1](a) f(x) = x sin1x(b) f(x) = e1x(c) f(x) = ln x(d) f(x) = cot x(e) f(x) = cos x cosx(f) f(x) = excos1x8. Surveyuniformlycontinuityofthefollowingfunctionon[0, +)(a) f(x) = x(b) f(x) = x sin x(c) f(x) = sin(x2)(d) f(x) = sin(sin x)(e) f(x) = sinx(f) f(x) = sin(x sin x)(g) f(x) = esin(x2)(h) f(x) = exarctan x9. ProvethatDirichletfunction.f(x) =_1 ifx Q0 ifx Iisntcontinuousatanypoint.10. Forexampleacontinuousfunctionandboundedon(0, 1]butthisfunctionisntuniformlycontinuous.11. Findtof(x) = xisuniformlycontinuouson[0, +).12. Givenf is continuous on[a, +) andthere limx+f(x) is finite. Prove f isboundedanduniformlycontinuouson[a, +).13. Givenf, gareuniformlycontinuousonA.Provethat(a) f+ g,f gareuniformlycontinuousonA.(b) f gisuniformlycontinuousonAiff, gareboundedonA.(c) Assertion(b)isnottruewhenoneof thetwofunctionsarenotbounded.Giveexamplesofthiscase.14. Showthateveryoddorderpolynomialalwaysexistatleastonerealsolution.15. Letf, garecontinuouson[a, b]satisfiesf(a) < g(a)andf(b) > g(b).Provethatf(x) = g(x)hassolutionon[a, b].16. Letf: [0, 1] [0, 1]iscontinuous.Showthatf(x) = xhassolutionon[0, 1].517. Letf(x) = xn+ an1xn1+ + a0withneven.Provethatthereexistsysuchthatf(y) f(x)forallx.18. Letf, g: [0, 1] [0, 1]aresurjectiveandcontinuous.Provethatthereexistsx0suchthatf(g(x0)) = g(f(x0)).6Differentialcalculus1 TheDerivativeoffunctions1. Computethederivative(ifthereexist)ofthefollowingfunctions.(a) f(x) = x|x|forx R.(b) f(x) =_|x|forx R.(c) f(x) = ln |x|forx R\ {0}.(d) f(x) = arccos1|x|for |x| > 1.(e) f(x) = logx 2for x > 0and x = 1.(f) f(x) =_x2sin1xifx = 0,0 ifx = 0.(g) f(x) = x cos1xvix = 0.(h) y= arctan(ln x).(i) y= etcos(t).(j) y= arcsin x2+ arccos x2.(k) y= xx.(l) y= xsin x.(m) y=_1 +1x_x.(n) y=xx.(o) y= xxx.2. Showthatthefunctiony= xexsatisfiestheequationxy

= (1 x)y.3. Showthatthefunctiony=11+x+ln xsatisfiestheequationxy

= (y ln x 1)y.4. Showthatthefunctiony= xex22satisfiestheequationxy

= (1 x2)y.5. Computethederivativeofimplicitfunctions.(a)_x = 2t 1y= t3(b)_x = a cos2ty= b sin2t(c)_x =1t+1y= (tt+1)2(d)_x =2at1+t2y=a(1t2)1+t2witha = 0.(e) x3+ y3= a3.(f) x3+ xy + y2= 0.(g) ln x + eyx= c.(h) ey= x + y.6. Findthederivativex

(y)if(a) y= 3x + x2.(b) y= x 12 sin x.(c) y= x + ex2.(d) y=5x132x.Aftercomputex

(y)aty= 1.7. Let_x = a(t sin t)y= a(1 cos t)(a > 0).Calculatey

(x)whent =2.8. Findy

(x)atindicatedpoints.7(a) (x + y)3= 27(x y)atM(2,1).(b) yey= ex+1atM(0,1).(c) y2= x + lnyxatM(1,1).9. Considerdifferentiableofthefollowingfunctions.(a) f(x) =_arctan x if |x| 1,4 sgn x +x12if |x| > 1.(b) f(x) =_x2ex2if |x| 1,1eif |x| > 1.10. Finda, b, c, dtofunctionfisdifferentiableon R.f(x) =___4x ifx 0,ax2+ bx + c if0 < x < 1,3 2x ifx 1.11. Define the domain which exist the inverse function x = x(y) of function y= x+ex.Calculatex

(y).12. Findthelimitofthefollowingsequences.(a) limx0(1 + x)5(1 + 5x)x2+ x5(b) limx0(1 + mx)n(1 + nx)mx2(c) limx(x + 1) (x2+ 1) ... (xn+ 1)[(nx)n+ 1]n+12(d) limx0n1 + x n1 + xx(e) limx0n1 + xm1 + x 1x(f) limxsin mxsin nx(g) limx3tan3x 3 tan xcos_x +6_(h) limx+ln (2 + e3x)ln (3 + e2x)(i) limx+x1x.(j) limx1x11x.(k) limx0+xsin x.(l) limx0x2sin1xsin x.(m) limxx sin xx + sin x.13. Provethatthefollowingfunctionsistwicedifferentiable.(a) f(x) =_e1x2ifx = 0,0 ifx = 0.(b) f(x) =_e1xifx > 0,0 ifx 0.8(c) f(x) =_e1xa+1xbifx (a, b),0 ifx/ (a, b).14. Computey

(x)ofthefollowingfunction.(a)_x = ln ty= t2(b)_x = cos 2ty= sin2t15. Findy

atpoint(1, 1)ifx2+ 5xy + y22x + y 6 = 0.16. Findy

atpoint(0, 1)ifx4xy + y4= 1.17. Findy(n)(x)offunction_x = ln ty= tm.18. Calculatednfofthefollowingfunctions.(a) f(x) =2x + 1x 3(b) f(x) =3x 1x2+ 3x 4(c) f(x) =x33x2+ 5x 73x2+ x 4(d) f(x) = ln(x23x + 2)2 Propertiesofdifferentialcalculus1. Exercises2,4,5,6,9,11inbookofA.Zorich,frompage232topage234.2. Exercises 325, 326, 327, 328, 329, 343, 344, 345, 346, 351, 355, 356, 359 in book ofexercises of Tran Duc Long, Nguyen Dinh Sang, Nguyen Viet Trieu Tien, HoangQuocToan.3. Letf: [a, +) Rhasderivativeofuptoorder2satisfiesi/ limx+f(x) = L.ii/ |f

(x)| Mforallx [a, +).Provethat limx+f

(x) = 0.3 Applicationofdifferentialcalculus1. Exercises1,2,3,5,6,8inbookofA.Zorich,frompage261topage263.2. ProveinequalitiesCachyandBunhiacopskibyusingconvexfunction.9