Question #8d080

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Answer 1 · 2017-06-11T22:29:35

#lim_(n->oo) (1+2/n)^(4n)= e^8#

Consider the sequence:

#a_n = ln ((1+2/n)^(4n)) = 4nln(1+2/n) = 8 (ln(1+2/n)/(2/n))#

Now we have :

#lim_(x->0) ln(1+x)/x = 1#

which means that the limit must be the same for any succession #{x_n}# such that #lim_(n->oo) x_n = 0#, so if we pose #x_n = 2/n# we have:

#lim_(n->oo) (ln(1+2/n)/(2/n)) = 1#

and then:

#lim_(n->oo) a_n = 8#

and since #e^x# is a continuous function in all of #RR#:

#lim_(n->oo) e^(a_n) = e^8#

Then:

#e^(a_n) = e^ln ((1+2/n)^(4n)) = (1+2/n)^(4n)#

#lim_(n->oo) (1+2/n)^(4n)= e^8#