Question

求下列极限谢谢各位

绿色框框里的





Answer 1

lim(x->0) sin(5x)/(6x)=lim(x->0) (5x)/(6x)=5/6

lim(x->0) sin(2x)/sin(5x)=lim(x->0) (2x)/(5x)=2/5
lim(x->0) (1-2x)^(1/x)=lim(x->0) {(1-2x)^[-1/(2x)]}^(-2)=e^(-2)
lim(x->∞) [(2x+3)/(2x+1)]^x=lim(x->∞) {[1+2/(2x+1)]^[(2x+1)/2]}^[2x/(2x+1)]=e
lim(x->∞) (1+5/x)^(-x)=lim(x->∞) [(1+5/x)^(x/5)]^(-5)=e^(-5)
lim(x->∞) [x/(x-1)]^(x-3)=lim(x->∞) {[1+1/(x-1)]^(x-1)}^[(x-3)/(x-1)]=e

Answer 2

(1)
let y=2x
lim(x->0) (1-2x)^(1/x)
=lim(y->0) (1-y)^(2/y)
=e^(-2)
(2)
(2x+3)/(2x+1) =1 + 2/(2x+1)
let
1/y =2/(2x+1)
lim(x->∞) [(2x+3)/(2x+1)]^x
=lim(x->∞) [1+ 2/(2x+1)]^x
=lim(y->∞) [1+ 1/y]^[(2y-1)/2]
=lim(y->∞) [1+ 1/y]^y
=e
(3)
let
5/x = 1/y
lim(x->∞) (1+ 5/x)^(-x)
=lim(y->∞) (1+ 1/y)^(-5y)
=e^(-5)
(4)
x/(x-1) = 1+ 1/(x-1)
let
1/y = 1/(x-1)
lim(x->∞) [x/(x-1)]^(x-3)
=lim(x->∞) [x/(x-1)]^x
=lim(x->∞) [1+ 1/(x-1)]^x
=lim(y->∞) [1+ 1/y]^(y+1)
=lim(y->∞) [1+ 1/y]^y
=e
(1)

lim(x->0) sin(5x)/(6x)
=lim(x->0) (5x)/(6x)
=5/6
(2)
lim(x->0) sin(2x)/sin(5x)
=lim(x->0) (2x)/(5x)
=2/5

Answer 3

Answer 4

如下

如上图