Calculating the Limit of (1+x)^(1/x) as x → ∞

Find \( \lim_{{x \to \infty}} (1 + x)^{1/x} \):

1. Rewrite with \( e \): \( \lim_{{x \to \infty}} e^{\ln((1+x)^{1/x})} \)
2. Put limit in exponent because \( e^u \) is continuous: \( e^{\lim_{{x \to \infty}} (\ln(1+x)/x)} \)
3. Now work on the limit:
   1. The limit \( \lim_{{x \to \infty}} \frac{\ln(1+x)}{x} \) has the form \( \frac{\infty}{\infty} \).
   2. Apply L’Hôpital’s Rule: \( \lim_{{x \to \infty}} \frac{1/(1+x)}{1} \).
   3. Simplify: \( \lim_{{x \to \infty}} \frac{1}{1+x} \).
   4. Take the limit: \( \frac{1}{1+\infty} = 0 \).
4. So we get \( e^0 = 1 \).