高数7道选择题，第一张是判断奇偶性

Question

有老师能给一下解题过程么，我会追加费用的

Answer 1

ln(-x√(1+(-x)^2))
=ln(1/(x+√(1+x^2)))
=-ln(x+√(1+x^2))
所以ln(x+√(1+x^2))是奇函数
.
-xarccos(-x/(1+(-x)^2))
=-xarccos(-x/(1+x^2))
=-x(π-arccos(x/(1+x^2)))
所以xarccos(x/(1+x^2))没有奇偶性
.
2.
lim[x→0+]exp(1/x)=+∞
lim[x→0-]exp(1/x)=0
所以选D

4.
lim[x→0+]xsin(1/x)=0
lim[x→0+]exp(1/x)=+∞
lim[x→0+]lnx=-∞
lim[x→0+]ln(1/x)=+∞
所以选A
.
2.
lim[]((x^2+1)/(1+x)-ax-b)
(x^2+1)/(1+x)-ax-b
=x-1+2/(1+x)-ax-b
=(1-a)x-1-b+2/(1+x)
所以由lim[x→∞]((x^2+1)/(1+x)-ax-b)=1
得：-1-b=1
即：b=-2
所以选B
.
3.
sinx无极限
lim[x→+∞]exp(-x)=0
lim[x→-∞]exp(-x)=+∞
所以lim[x→∞]exp(-x)无极限
lim[x→∞]((x+1)/(x^2+1))=lim[x→∞](1/(x-1))=0
lim[x→+∞]arctanx=π/2
lim[x→-∞]arctanx=-π/2
所以lim[x→∞]arctanx无极限
选择C
.
4.
x^3/(x^2+1)-x^2/(x-1)
=(x/(x^2+1)-1/(x-1))x^2
=-(x+1)x^2/(x^2+1)/(x-1)
=-(1+1/x)/(1+1/x^2)/(1-1/x)
所以lim[x→∞](x^3/(x^2+1)-x^2/(x-1))=-1
.
5.
lim[x→∞]((3x^2-1)/(x^3+2x+1))
=lim[x→∞]((3-1/x^2)/(1+2/x^2+1/x^3)/x)
=0
所以选A

Answer 2