How to properly utilize the definition of e for a specific limit

Question

I have tried to solve the following limit, but I'm stumped.

$\lim_{n \to +\infty} (2-\frac{3}{n})^n$

Is there an effective way to rewrite it? So far, I've rewritten it into

$\lim_{p \to +\infty} ((1-\frac{1}{p})^p)\frac{3}{p-1}$

Where $p=\frac{n}{n-3}$

But what then?

Answer 1

You can apply the exponential function and the natural logarithm to the general expression as follows: $$\mathrm{e}^{\ln\left(2-\frac{3}{n}\right) ^n},$$ and use the logarithm properties to get the expression $$\lim_{ n \to \infty} \mathrm{e}^{n\ln \left(\frac{(2n-3)}{n}\right)},$$ which diverges positively.