例一：1lim(3n232n233n2...3(n1)n2)nn2解析：13232323232lim2(n2n3n...(n1)nn*nn)nn1nlim(3n232n233n2...3(n1)n23n*n2)limnnnn221lim(3312331...3nn3)0nnnnnnn1n13i133lim*xdxnnn0i14例二：1lim1[(2cosx)x1]x0x3解析：1ln2cosxlim1[ex31]x0x1ln2cosx-1(又因为lim1ex3=lim1ln（1+cosxx-1）=lim1cos-1=lim20)x0xx0x3x0x3x03x112cosx1cosx-1cosx-11limln()lim22ln（1+）limx0xx3x0xx3x036变形样式二：1例三：tan(tanxx)sin(sin)limx0tanxxsin解析：tan(tanx)tan(sinx)tan(sinx)sin(sinx)limtanxsinxlimtanxsinxxx002tan(sinxx)(1cos(sin))limsecxlimtanxx(1cos)xx0012sinxx2sin1lim12x0xx22例四：设p(x)dn(1xmn),其中m,n为两正整数，则p(1)________。dxn解析：因(1xm)n(1x)n(1xx21...xm)n=uvnp(x)=dn(1xm)n(uv)nckukvnkdxnnk0nkknknnnknnp(1)cnnu1v1cu1v1(1)n!mk02例五：设函数fx()在x=a处可导，则函数|f(x)|在x=a处不可导的充分条件是_________。A,f(a)=0且f''(a)=0B,f(a)>0且f(a)>0C,f(a)<0且f''(a)<0D,f(a)=0且f(a)0解析：分析A,取f(x)=|x|在x=0处可导，f(0)=0,f'(0)=0,|f(x)|=|x|x||=x2在x0处可导分析B,f()a0,limf()xf()a0xa由极限保号性知u(a,)(a,a)当x(a,)时，有f(x)0,lim|()||()|fxfalimf()xf()af'(a)xaxaxaxa分析C,()0,lim()fafxf()0a由极限保号性知u(,)(aa,a)xa当x(a,)时，有f(x)0,lim|()||()|fxfalimf()()xfaf'(a)xaxaxaxa分析D（正确选项），f(a)0,f'(a)01,f''(a)0,0f(a)limf()()()xfalimfxxaxaxaxa由极限的保号性去心领域va(,)fx()当x去心领域v(a,)有xa0|f(x)||f(x)|f(x)'当x(a,a)时，f(x)0,limxalimxaf(a)xaxa|f(x)||f(x)|f(x)'当x(a,a+)时，f(x)0,limxalimxaf(a)xaxa变形样式一：变形样式二：3例六：设fx()是可导偶函数，则下列函数中比为奇函数的是（）xA,f()td1txB,tf(1t2)d0txC,t2f'(t)d0tD,()x23fx解析：1、可导奇函数，其导函数必为偶函数可导偶函数，其导函数必为奇函数可导周期函数，其导函数必为周期函数x2，f(x)为连续奇函数，则fd(t)为可导偶函数atxf(x)为连续偶函数，则f()td为可导奇函数0txf(x)为连续T周期函数则f()td为周期函数atTf(x)d(x)00由以上知：f()()x33fx例七：4设在x=0的某邻域内，函数f(x)满足f''(x)+[f(x)]'22sinx,则下列命题满足：A，当f(x)'0时，x=0不是f(x)极值点，（0，f(0))是曲线y=f(x)的拐点；B，当f(x)'=0时，x=0是fx()极值