limit as x approaches infinity of ((x+3)/(x+1))^x?

The definition of e^r = lim n->inf (1 + r/n)^n

((x + 3) / (x + 1))^x =>

((x + 1 + 2) / (x + 1))^x =>

((x + 1) / (x + 1) + 2 / (x + 1))^x =>

(1 + 2/(x + 1))^x =>

(1 + 2/(x + 1))^(x + 1 - 1) =>

(1 + 2/(x + 1))^(x + 1) / (1 + 2/(x + 1))

x goes to infinity, which means that x + 1 goes to inf + 1, which is just infinity. If we let x + 1 = t

(1 + 2/t)^t / (1 + 2/t)

t goes to infinity

By our original definition, the numerator is e^2

e^2 / (1 + 1/inf) =>

e^2 / (1 + 0) =>

e^2 / 1 =>

e^2

L'H does work...rewrite .... x [ ln (x+3) - ln (x+1) ] = [ ln (x+3) - ln(x+1) ] / [ 1/x].....0/0..

L'H : [ 1 / (x+3) - 1/(x+1) ] / [ -1/x² ] = [ x² ] [ 2 / (x+3)(x+1) ] ----> 2.....thus e²

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