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

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Answer 1 · 2016-01-21T07:05:52

#1/e^3#

Let #L = lim_(x->oo)(1-3/x)^x#

#=> ln(L) = ln(lim_(x->oo)(1-3/x)^x)#

#=lim_(x->oo)ln((1-3/x)^x)# (because #ln# is continuous)

#=lim_(x->oo)xln(1-3/x)#

#=lim_(x->0)ln(1-3x)/x# (note the change of limit)

#=lim_(x->0)(d/dxln(1-3x))/(d/dxx)# (#0/0# L'hospital case)

#=lim_(x->0)3/(3x-1)#

#= 3/(0-1)#

#= -3#

So, as #ln(L) = -3#, we can exponentiate to obtain

#L = e^ln(L) = e^(-3) = 1/e^3#