6道简单极限题

1,
lim(x->2/PI)(sinx)^(tanx) = sin(2/PI)^[tan(2/PI)]

2,
lim(x->0)[1+3(tanx)^2]^[(cotx)^2] = lim(x->0){[1+3(tanx)^2]^[3(cotx)^2]}^(1/3)
= e^(1/3)

3,
lim(x->PI)[1-tanx]^[1/sin(2x)] = lim(x->PI){[1-tanx]^[-cotx]}^[-1/(2(cosx)^2)]
= e^[-1/2]

4,
lim(x->0)[(a^x+b^x)/2]^(1/x)
lim(x->0)[ln[a^x+b^x]-ln2]/x = lim(x->0)[(a^xlna+b^xlnb)/(a^x+b^x)]/1 = (lna+lnb)/2,
lim(x->0)[(a^x+b^x)/2]^(1/x)
= e^{lim(x->0)[ln[a^x+b^x]-ln2]/x}
= e^[(lna+lnb)/2]
= e^[ln[(ab)^(1/2)]]
= (ab)^(1/2)

5,
lim(x->inf)arctanx/x = 0

6,
lim(x->-inf)[(x^2+2x)^(1/2)-(x^2-x)^(1/2)] = lim(x->-inf)[3x]/[x^2+2x)^(1/2)+(x^2-x)^(1/2)]
= lim(x->-inf)[-3]/[(1+2/x)^(1/2)+(1-1/x)^(1/2)]
= -3/[1+1] = -3/2

1.
直接就是把x带进去 得(sin2/π)^tan2/π
2.换元法令t=cot^2x (t>0)
利用 重要极限那个 等于e的
可得原式=e^3
3第三题
换为E的指数
原式=e^ln(1-tanx)/sin2x