What is the exact limit?

Question

Find the limit exactly.
#lim_(xrarr∞)(1+3/n)^n#

1971 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-11-10T00:03:53

#lim_(n->oo) (1+3/n)^n = e^3#

We can start from the limit:

#lim_(x->oo) (1+1/x)^x = e#

Consider now:

#lim_(x->oo) (1+3/x)^x #

and substitute #x=3y# to have:

#lim_(x->oo) (1+3/x)^x = lim_(y->oo) (1+3/(3y))^(3y) = lim_(y->oo) ((1+1/y)^y)^3 = e^3#

Then the sequence we obtain in the particular case where #y=n# converges to the same limit.