1. Start with the expression you want to find the limit of:

\(x^{1/x}\)

This expression represents \(x\) raised to the reciprocal of itself, and we want to find its behavior as \(x\) approaches 0.

2. Rewrite the expression using \(e\) as the base:

\(e^{(\ln(x^{1/x}))}\)

By using the logarithmic property \(\ln(a^b) = b \ln(a)\), we can rewrite the expression in terms of \(e\), which will allow us to apply the limit more easily:

\(e^{((1/x) \ln(x))}\)

3. Apply the limit to the exponent:

\(\lim_{x \to 0} (1/x) \ln(x)\)

This limit is an indeterminate form (0/0), so we can use L’Hôpital’s Rule to differentiate the numerator and denominator:

\(\lim_{x \to 0} \frac{\ln(x)}{-1/x^2} = \lim_{x \to 0} \frac{1/x}{2/x^3} = \lim_{x \to 0} \frac{x^2}{2} = 0\)

The limit of the exponent is 0, which simplifies our expression.

4. Substitute the limit back into the original expression:

\(e^0 = 1\)

Since the base \(e\) raised to the power of 0 is 1, the limit of the original expression is 1.

So the limit of \(x^{1/x}\) as \(x\) goes to 0 is 1.