Question

求这四题过程和答案

Answer 1

（5）原式=lim(x->+∞) [(1+2/x)^(x/2)]^[2(x+1)/x]
=e^lim(x->+∞) (2+2/x)
=e^2
（6）原式=lim(x->+∞) 2lnx*(1/x)/3x^2
=(2/3)*lim(x->+∞) lnx/x^3
=(2/3)*lim(x->+∞) (1/x)/3x^2
=0
（7）原式=lim(x->0) (-sin3x)/x
=lim(x->0) (-3x)/x
=-3
（8）原式=lim(x->-∞) [-e^(-x)]/2x
=lim(x->-∞) e^(-x)/2
=+∞

Answer 2

如图

Answer 3

(5)
lim(x->∞) [(x+2)/x]^(x+1)
=lim(x->∞) (1+ 2/x)^(x+1)
=lim(x->∞) (1+ 2/x)^x
=e^2
(6)
lim(x->∞) (lnx)^2/x^3 (∞/∞)
=lim(x->∞) [2(lnx)/x]/(3x^2)
=(2/3) lim(x->∞) lnx/x^3 (∞/∞)
=(2/3) lim(x->∞) (1/x)/(3x^2)
=(2/9) lim(x->∞) 1/x^3
=0
(7)
lim(x->0) ln(1-sin3x)/ln(1+x)
=lim(x->0) ln(1-3x)/ln(1+x)
=lim(x->0) -3x/x
=-3
(7)
lim(x->∞) (1+e^(-x))/(1+x^2)

=lim(x->∞) (e^x+1)/ [e^x.(1+x^2)] (∞/∞)
=lim(x->∞) e^x/ [e^x.(1+x^2+2x)]
=lim(x->∞) 1/ (1+x^2+2x)
=0