Approach to Solution:
To find the value of the limit as x approaches infinity for the given expression, we can utilize the properties of the gamma function and apply L'Hôpital's Rule.

Applying L'Hôpital's Rule:
1. Rewrite the expression in terms of the gamma function:
γ(x+1) / (x/e)^x * √x = Γ(x+1) / (x/e)^x * x^(1/2)
2. Take the natural logarithm of the expression to simplify the calculation:
ln(Γ(x+1) / (x/e)^x * x^(1/2))
3. Apply L'Hôpital's Rule by taking the derivative of the numerator and denominator separately:
lim x→∞ ln(Γ(x+1) / (x/e)^x * x^(1/2))
= lim x→∞ [ln(Γ(x+1)) - x*ln(x/e) + 1/2*ln(x)]
4. Evaluate the limit of the derivative:
= lim x→∞ [digamma(x+1) - (ln(x) + 1) + 1/(2x)]

Conclusion:
By simplifying the expression and applying L'Hôpital's Rule, the value of the limit as x approaches infinity for the given expression is not a simple value like √2π, 1/2, 1/2π, or √π/2. It involves the digamma function and logarithmic terms, which do not lead to a straightforward numerical result.