Answer
$$1$$
Work Step by Step
Definition of a sequence defined by a function: Let us consider a function $f(x)$ whose limit $\lim\limits_{x \to \infty} f(x)$ exists then the sequence $a_n=f(n)$ will converge to the same limit.
That is, $\lim\limits_{n \to \infty} a_n=\lim\limits_{x \to \infty} f(x)$
Here, in this problem we have $a_n=(\dfrac{1}{n})^{1/n}$ and $f(x)=(\dfrac{1}{x})^{1/x}$
Next, $\lim\limits_{n \to \infty} a_n=\lim\limits_{x \to \infty} (\dfrac{1}{x})^{1/x}$
or, $\lim\limits_{n \to \infty} a_n=\lim\limits_{x \to \infty} \dfrac{1}{x^{1/x}}=\lim\limits_{x \to \infty} \dfrac{1}{e^{(1/x) \ln x}}$
or, $\lim\limits_{n \to \infty} a_n= \dfrac{1}{e^{\lim\limits_{x \to \infty}(1/x) \times \lim\limits_{x \to \infty} \ln (x)}}$
Since, $n \gt 0$ for all the values of $n$ and $\lim\limits_{n \to \infty} e^n=\infty$ and $\lim\limits_{n \to \infty} e^{-n}=0$
Thus. $\lim\limits_{n \to \infty} a_n=\dfrac{1}{e^0}=1$