What is the limit of # ((1)+(3/x)+(5/x^2))^x# as x approaches infinity?

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Answer 1 · 2016-08-07T17:34:34

#e^3#

# ((1)+(3/x)+(5/x^2))^x = (1+(3+5/x)/x)^x#

substituting #y = x/3# we have

# (1+(3+5/x)/x)^x equiv (1+(3+5/(3y))/(3y))^{3y} = #
#(1+(1+5/(9y))/y)^{3y} =( (1+(1+5/(9y))/y)^y)^3#

then

#lim_{x->oo} (1+(3+5/x)/x)^x = lim_{y->oo}( (1+(1+5/(9y))/y)^y)^3 = #
#(lim_{y->oo} (1+(1+5/(9y))/y)^y)^3 = e^3#

Note. We used the textbook result.

#lim_{y->oo} (1+1/y)^y = e#

Answer 2 · 2016-08-07T18:47:41

#e^3#.

Reqd. Limit #=lim_(xrarroo)(1+3/x+5/x^2)^x#

#=lim_(xrarroo)(1+(3x+5)/x^2)^x#

#=lim_(xrarroo){(1+(3x+5)/x^2)^(x^2/(3x+5))}^((3x+5)/x)#

#=lim_(xrarroo){(1+(3x+5)/x^2)^(x^2/(3x+5))}^((3+5/x)#

#=lim_(xrarroo){(1+(3x+5)/x^2)^(x^2/(3x+5))}^3*{(1+(3x+5)/x^2)^(x^2/(3x+5))}^(5/x)#

#=e^3*e^0#

#=e^3#, as Cesareo R., Sir has derived!

Enjoy Maths!