Question

Does `a_n={(3/n)^(1/n)}`converge? If so what is the limit?

Answer 1

It converges to `1`.

Explanation:

`f(x) = (3/x)^(1/x) = e^(ln(3/x)/x)`

`lim_(xrarroo)(ln(3/x)/x)` has initial form `(-oo)/oo` which is indeterminate, so use l'Hospital's rule.

`lim_(xrarroo) (x/3(-3/x^2))/1 = lim_(xrarr0)-1/x = 0`

Since the exponent goes to `0`, we have

`lim_(xrarroo)(ln(3/x)/x) = e^0 = 1`