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

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Answer 1 · 2016-08-08T13:24:31

1

#lim_(x to oo) (1+2x)^(1/x)#

#= lim_(x to oo) e^ ( ln (1+2x)^(1/x) )#

#=e^ ( lim_(x to oo) ln (1+2x)^(1/x) )# as exponential function is continous

#=e^L #

#L = lim_(x to oo) ln (1+2x)^(1/x) #

# = lim_(x to oo) 1/xln (1+2x) #

# = lim_(x to oo) (ln (1+2x))/x #

which is # oo/oo # indeterminate so we can use L Hopital

# = lim_(x to oo) (1/ (1+2x))/1 #

#implies L = lim_(x to oo) 1/ (1+2x) = 0#

and
#e^0 = 1#