How to find $\lim_{x\rightarrow\infty}\bigl(\frac{x+1}{x-2}\bigr)^{2x-1}$ $$\lim_{x\rightarrow\infty}\left(\frac{x+1}{x-2}\right)^{2x-1}$$
What are the steps to solve it? Probably the division should be multiplied by some expression.

$$\left(\frac{1+x}{-2+x}\right)^{-1+2x}=\left[\left(1+\frac3{x-2}\right)^{x-2}\right]^{2}\left(1+\frac3{x-2}\right)^3$$
And now use the basic lemma
$$\lim_{x\to\infty}\left(1+\frac a{f(x)}\right)^{f(x)}=e^a$$
where $\;a\in\Bbb R\;$ and $\;f\;$ is a funcion s.t.
$$\lim_{x\to\infty}f(x)=\infty$$
