calculus - Evaluating $limlimits_{n to infty} left(1+{3 over n}right)^n$

$$\lim_{n \to \infty} \left(1+{3 \over n}\right)^n$$

What are the general rules for limit of this kind, like $\lim_{n \to \infty} \left(1+{\alpha \over n}\right)^n$ or $\lim_{n \to \infty} \left(1+{\alpha \over \beta n}\right)^n$

And how can I solve this?

Answer

Notice, we know that $$\lim_{n\to \infty}\left(1+\frac{1}{n}\right)^n=e$$

General rule: let $$\frac{\alpha}{\beta n}=\frac{1}{t}\implies n=\frac{\alpha }{\beta }t$$ hence, we get $$\lim_{n\to \infty}\left(1+\frac{\alpha}{\beta n}\right)^n=\lim_{t\to \infty}\left(1+\frac{1}{t}\right)^{\frac{\alpha }{\beta }t}$$ $$=\left(\lim_{t\to \infty}\left(1+\frac{1}{t}\right)^{t}\right)^{\frac{\alpha }{\beta }}$$ $$=\large{e^{\frac{\alpha }{\beta }}}$$