# Find $\lim\limits_{n \to \infty}\left(\frac{\alpha}{x/n + \alpha}\right)^{\beta n}$.

Find $$\lim\limits_{n \to \infty}\left(\frac{\alpha}{x/n + \alpha}\right)^{\beta n}$$ where $\alpha, \beta, x > 0$.

I have a solution here that says that it is $e^{-x\beta / \alpha}$, but it isn't easy for me to see this from the limit definition: we would have $$e^{-x\beta / \alpha} = \lim_{n \to \infty}\left(1+\dfrac{-x\beta/\alpha}{n} \right)^{n}\text{.}$$ Is there a fancy substitution that I'm not seeing that shows that these two forms are equivalent?

Note that this was for a timed qualifying exam, so efficient solutions are preferable.

One may write $$\left(\frac{\alpha}{x/n +\alpha}\right)^{\beta n}=\left[\left(1+\frac{x/\alpha}{n} \right)^n\right]^{-\beta}$$ then let $n \to \infty$.