以下趨於多少均省略，注意加上：

（5）lim2^n*sinx/2^n

＝lim x*sin(x/2^n)/[x/2^n]

=xlimsin(x/2^n)/(x/2^n)

=x*1=x

(7　)lim[sqr(1-cosx)]/x =lim根號2 *sin(x/2) /x

＝sqr2 *limsin(x/2)/(x/2) /2

=(根號2）/2

（1）lim(1+3/n)^(-n)

=lim[(1+1/(n/3))^(n/3)]^(-3)

=e^(-3)

(3) lim [(x+3)/(x+1)]^x

=lim[1+2/(x+1)]^x

=lim{[1+1/((x+1)/2)]^[(x+1)/2]}^2 / lim[1+2/(x+1)]

=e^2 /1

=e^2

(5)lim(1-2x)^(1/x)

=lim{{1+1/(-1/(2x)]}^(-1/(2x))}^(-2)

=e^(-2)

(7) lim(1+tanx)^cotx

=lim(1+1/cotx)^cotx

=e

(1)limxsin(1/x)

=limsin(1/x)/(1/x)

=1

(2)limsin(ax)/sin(bx)

分子分母同時趨於0，用諾貝塔法則，分子分母同時求導數：

＝limacos(ax)/(bcos(bx))

=a*1/(b*1)

=a/b