Can You Help Me Compute These Limits as X Approaches Infinity?

[MathematicalPhysicist]

i need to compute the lim of (1+2/x)^x or (1-1/x)^x and (1+1/x)^(x+3) when x approaches infinity.
if you can provide a steady method to compute it, it will be appreciated.

btw i know that lim (1+1/x)^x as x->inf is "e", but does it have any correlation to here.

 

[ehild]

i need to compute the lim of (1+2/x)^x or (1-1/x)^x and (1+1/x)^(x+3) when x approaches infinity.
if you can provide a steady method to compute it, it will be appreciated.
btw i know that lim (1+1/x)^x as x->inf is "e", but does it have any correlation to here.

Yes, it does.
For the first problem, replace x/2=y. You should determine lim (1+1/y)^(2y) as y->infinity.
lim (1+1/y)^(2y)=lim ((1+1/y)^2)->
(lim (1+1/y))^2 = e^2 if y->infinity. But this is equivalent with the original limit when x -> infinity.
As for the second problem, (1+1/x)^(x+3)=((1+1/x)^x)*(1+1/x)^3. You can proceed from here.
In case of the third problem
1-1/x=1/[1+1/(x-1)]
Let be y= x-1. If x -> infinity , so does y.
You have to detemine the limit
lim (1-1/x)^x when x->infinity. It is equivalent with
lim[(1/(1+1/y)]^(y+1)) when y->infinity.
lim[(1/(1+1/y)]^(y+1))=
1/lim[(1+1/y)^(y+1)]=1/lim[(1+1/y)^y*(1+1/y)] =1/[lim(1+1/y)^y*lim(1+1/y)]=1/e
ehild

 

[neurocomp2003]

arrange and factorize

 

[MathematicalPhysicist]

thanks, i think it's a simple substitution.