12007–2008�1���5p���6���K�!O�e��K(zK5�,�35�)1.limn→∞��n+3√n−�n−√n�;2.limx→+∞√x+√x√1+x;3.limx→0�1x−cosxex−1�;4.�e√xdx;5.�30x√x+1dx;6.�e1xlnxdx;7.�dxex+e−x.)1.limn→∞��n+3√n−�n−√n�=limn→∞n+3√n−(n−√n)√n+3√n+√n+√n=limn→∞4√n√n+3√n+√n+√n=2.2.limx→+∞√x+√x√1+x=1.3.limx→0�1x−cosxex−1�=limx→0ex−1−xcosxx(ex−1)=limx→0ex−1−xcosxx2=limx→0ex−cosx+xsinx2x=limx→012(ex+2sinx+xcosx)=12.4.�e√xdx=�et·2tdt=2�tdet=2�tet−�etdt�=2et(t−1)+C=2e√x(√x−1)+C.5.�30x√x+1dx=�30(x+1)−1√x+1d(x+1)=�30�√x+1−(x+1)−12�d(x+1)=�23(x+1)32−2(x+1)12����30=83.6.�e1xlnxdx=�e1lnxd�x22�=x22lnx���e1−�e1x22·1xdx=e22−�e1x2dx=e22−�e24−14�=e2+14.7.�dxex+e−x=�dex(ex)2+1=arctanex+C(�−arctane−x+C)�!)�e��K�zK7�,�35��1.I=limx→∞�sin1x+cos1x�x.){�I=limx→∞�sin1x+cos1x�x=limx→0(sinx+cosx)1x=elimx→0ln(sinx+cosx)x=elimx→0cosx−sinxsinx+cosx=e.2){�I=limx→∞�sin1x+cos1x�x=elimx→∞ln(sin1x+cos1x)1x=elimx→∞(cos1x−sin1x)(−1x2)−1x2=e){nI=limx→∞��sin1x+cos1x��x2=limx→∞�1+sin2x�x2=elimx→∞x2·sin2x=e2.a,b����f(x)=�ax+bcosx,x≤0sinxx−x,x>03x=0��.)�f(0)=b,f(0+)=limx→0+�sinxx−x�=1,2df(x)3x=0?��7�Y�f(0)=f(0+).=b=1.l���/�Ǒf(x)=�ax+cosx,x≤0sinxx−x,x>0�f′−(0)=limx→0−(ax+cosx)−1x=a+limx→0−cosx−1x=a,f′+(0)=limx→0+sinxx−x−1x=limx→0+�sinx−xx2−1�=limx→0+sinx−xx2−1=limx→0+cosx−12x−1=−1,2df(x)3x=0?���f′−(0)=f′+(0).=a=−1.��,a=−1,b=1.3.d��exy=x+y(����y=y(x)3:(0,1)��!����.3)dK��,�x=0�,y=1.���exy=x+y�>����exy(y+xy′)=1+y′.-x=0k1=1+y′(0).=y′(0)=0.UY�x��exy(y+xy′)2+exy(2y′+xy′′)=y′′.-x=0Kk1+2y′(0)=y′′(0).=y′′(0)=1.4.y=f(lnx)ef(x)������f(x)��.)y′=f′(lnx)·1x·ef(x)+f(lnx)·ef(x)·f′(x)=ef(x)�f′(lnx)x+f(lnx)·f′(x)�.�d�dy=ef(x)�f′(lnx)x+f(lnx)·f′(x)�dx.5.�f(x)�Y��limx→0f(x)x=A,ϕ(x)=�10f(xt)dt,ϕ′(x)�?�ϕ′(x)3x=0��Y5.)�Ǒlimx→0f(x)x=A,��l...
