Inexistence or limit that does not exist

Question

We suppose that have this limit:

$$\lim _{x\to +\infty }\frac{(x-1)^{\sqrt x}}{x-2}$$

Are there theorems in Mathematical Analysis, corollaries that use successions, particular strategies, that help me to demonstrate that a limit exists or does not exist?

Related question: Limits that do not exist: search of general techniques

Answer 1

We have that

$$\frac{(x-1)^{\sqrt x}}{x-2}\ge \frac{(x-2)^{\sqrt x}}{x-2}=(x-2)^{\sqrt x-1}\to \infty$$

since it is in the form $\infty^\infty$.