1.原式=lim(x->1){[(1+(1-x))^(1/(1-x))]^[(1-x)/cot(πx/2)]}
={lim(x->1))[(1+(1-x))^(1/(1-x))]}^{lim(x->1)[(1-x)/cot(πx/2)]}
=e^{lim(x->1)[(1-x)/cot(πx/2)]} (应用lim(x->0)[(1+x)^(1/x)]=e)
=e^{lim(x->1)[-1/((-π/2)csc²(πx/2))]} (0/0型,应用罗比达法则)
=e^(2/π);
2.原式=lim(x->∞){[sinln(1+3/x)-sinln(1+1/x)]/(1/x)}
=lim(x->0){[sinln(1+3x)-sinln(1+x)]/x} (用x代换1/x)
=lim(x->0)[3cosln(1+3x)/(1+3x)-cosln(1+x)/(1+x)] (应用罗比达法则)
=3*1/(1+0)-1/(1+0)
=2;
3.原式=lim(x->0)[(e^tanx)(e^(x-tanx)-1)/(x-tanx))]
=lim(x->0)(e^tanx)*lim(x->0)(e^(x-tanx)-1)/(x-tanx))
=1*lim(x->0)(e^x-1)/x) (用x代换x-tanx)
=lim(x->0)(e^x) （0/0型,应用罗比达法则）
=1.