Evaluation of a limit involving gamma function

Question

Basically I try to answer this question. I am almost able to prove it straight on, but I hit a roadblock at the very last step. Namely, the limit $$ \lim_{n\rightarrow \infty} \left( \sqrt[n+1]{\Gamma \left( n+1+\frac{1}{2} \right)}-\sqrt[n]{\Gamma \left( n+\frac{1}{2} \right)} \right) $$ It should evaluate to $1/e$ through numerical estimation. One possible direction here is to justify that we just plug in Stirling's formula, due to $$ \sqrt[n]{\Gamma \left( n+\frac{1}{2} \right)}\sim \frac{n}{e} $$ we have $$ \lim_{n\rightarrow \infty} \left( \sqrt[n+1]{\Gamma \left( n+1+\frac{1}{2} \right)}-\sqrt[n]{\Gamma \left( n+\frac{1}{2} \right)} \right) =\lim_{n\rightarrow \infty} \left( \frac{n+1}{e}-\frac{n}{e} \right) =\frac{1}{e} $$

Answer 1

Hint: Stirling's approximation $$\ln \Gamma(t) = t\ln t - t -\frac12\ln t + \frac12 \ln(2\pi) + O\left(\frac1t\right)$$