\(lim_{𝑥→∞} \ \ 𝑥^{1/𝑥} \)

Key idea: rewrite the power using the exponential and natural logarithm to evaluate the limit of the exponent.

1. Rewrite the expression: x^{1/x} = exp((1/x) \u200bln x).

2. Evaluate the exponent: consider (ln x)/x as x → ∞. This ratio tends to 0 (for example, apply l'Hôpital's rule: derivative of ln x is 1/x and derivative of x is 1, so the limit is 0).

3. Conclude the limit: exp((ln x)/x) → exp(0) = 1, so the limit of x^{1/x} as x → ∞ is 1.

Therefore the limit equals 1.