I have been trying to solve the following limit but am completely stuck.
$$\lim_{\alpha \rightarrow \infty} 1-\left( \frac{y+\alpha}{\alpha-1} \right)^{-\alpha}$$
I have tried inverting the ratio and came up with the following expression:
$$ 1 - \lim_{\alpha \rightarrow \infty} \left( 1-\frac{y+1}{y+\alpha}\right)^\alpha$$
Which roughly resembles the exponential function:
$$\lim_{\alpha \rightarrow \infty} \left( 1- \frac{x}{\alpha} \right)^\alpha = \exp(-x)$$
Except for the additive term in the denominator. Is there a u-substitution type trick to this?