\[ \begin{split} &\lim_{t \to \infty}\left(\frac{t}{t+1}\right)^t =\\[6pt] &=\lim_{t \to \infty}\left(\frac{t+1-1}{t+1}\right)^t =\\[6pt] &=\lim_{t \to \infty}\left(1-\frac{1}{t+1}\right)^t =\\[6pt] &=\begin{vmatrix} -\dfrac{1}{t+1}=z; && t=-\dfrac{1}{z}-1;\\ t \to \infty &to& z \to 0 \end{vmatrix}=\\[6pt] &=\lim_{z \to 0}(1+z)^{-\dfrac{1}{z}-1}=\\[6pt] &=\lim_{z \to 0}\dfrac{\left[\left(1+z\right)^{\dfrac{1}{z}}\right]^{-1}}{1+z}=\\[6pt] &=\frac{e^{-1}}{1}=\frac{1}{e} \end{split} \]