Apply an exponential of a logarithm to the expression.
$\lim_{x \to \infty} x^{\frac{1}{x}}=\lim_{x \to \infty}\exp\left ( \log\left (x^{\frac{1}{x}} \right ) \right )=\lim_{x \to \infty}\exp\left ( \frac{\log\left ( x \right )}{x} \right )$
$\lim_{x \to \infty}\exp\left ( \frac{\log\left ( x \right )}{x} \right )$
Since the exponential function is continuous, we may factor it out of the limit.
$\lim_{x \to \infty}\exp\left ( \frac{\log\left ( x \right )}{x} \right )=\exp\left (\lim_{x \to \infty}\frac{\log\left ( x \right )}{x} \right )$
$\exp\left ( \lim_{x \to \infty}\frac{\log\left ( x \right )}{x} \right )$
Logarithmic functions grow asymptotically slower than polynomials.
Since $\log\left ( x \right )$ grows asymptotically slower than the polynomial $x$ as $x$ approaches $\infty$,
$\lim_{x \to \infty}\frac{\log\left ( x \right )}{x}=0$:
$e^{0} = 1$
Correct Answer: $C$