What is the limit of ((1)+(3/x)+(5/x^2))^x((1)+(3x)+(5x2))x as x approaches infinity?

2 Answers
Aug 7, 2016

e^3e3

Explanation:

((1)+(3/x)+(5/x^2))^x = (1+(3+5/x)/x)^x((1)+(3x)+(5x2))x=(1+3+5xx)x

substituting y = x/3y=x3 we have

(1+(3+5/x)/x)^x equiv (1+(3+5/(3y))/(3y))^{3y} = (1+3+5xx)x1+3+53y3y3y=
(1+(1+5/(9y))/y)^{3y} =( (1+(1+5/(9y))/y)^y)^31+1+59yy3y=1+1+59yyy3

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

Aug 7, 2016

e^3.

Explanation:

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!