[例3.23]设-|||-in (0,1), 证明-|||-(1) (1+x)(ln )^2(1+x)lt (x)^2;-|||-(2) dfrac (1)(ln 2)-1lt dfrac (1)(ln (1+x))-dfrac (1)(x)lt dfrac (1)(2)

证明:ln dfrac (1+x)(1-x)+cos xgeqslant 1+dfrac ({x)^2}(2) -1lt xlt 1.证明:.

计算：lim _(xarrow 0)dfrac ({(1+x))^dfrac (2{x)}-(e)^2(1-ln (1+x))}(x)。计算：。

计算极限 lim _(xarrow 0)dfrac ({(1+x))^dfrac (2{x)}-(e)^2[ 1-ln (1+x)] }(x) +x)]/

设 lim _(xarrow 0)dfrac (ln (1+x)-(ax+b{x)^2)}({x)^2}=2, 则设 lim _(xarrow 0)dfrac

求极限 lim _(xarrow 0)[ dfrac (1)(x)-dfrac (1)({x)^2}ln (1+x)] .

[题目]当 gt 0 时,证明: -dfrac ({x)^2}(2)lt ln (1+x)lt x.

求下列极限-|||-lim _(xarrow 0)[ dfrac (1)(ln (x+sqrt {1+{x)^2})}-dfrac (1)(ln (1+x))]

利用导数证明:当 gt 1 时, dfrac (ln (1+x))(ln x)gt dfrac (x)(1+x)

19.证明:当 gt 0 时, (1+x)(ln )^2(1+x)lt (x)^2.

[题目 lim _(xarrow 0)dfrac (ln {(1+x))^dfrac (1{x)-1}}(x)