求下列极限：
(1)limx→0√ex+2x+1；(2)limx→π/4ln(tanx)；
(3)limx→∞e(2x2-3)/(x2+1)(4)limx→α[(sinx-sinα)/(x-α)]；
(5)limx→∞[(x+1)/x]2x+1；(6)limx→0(1+3tan2)cot2x
(1)limx→0√ex+2x+1=√1+0+1=√2 (2)limx→π/4ln(tanx)=In(limx→π/4ln(tanx)tanx)=lnl=0 (3)limx→∞e(2x2-3)/(x2+1)=elimx→∞(2x2-3)/(x2+1)=e2 (4)limx→α[(sinx-sinα)/(x-α)]=limx→α{2sin[(x-α)/2]cos[(x+α)/2]}/(x-α) =limx→α[sin(x-α)/2]/[(x-α)/2]•limx→αcos[(x+α)/2]=cosα (5)limx→∞[(x+1)/x]2x+1=limx→∞(1+1/x)x](2x+1)x=e2 (6)limx→0(1+3tan2)cot2x=limx→0[(1+3tanx)1/3tan2x]3=e3
