Limit of $t^{p - 1}$ as $t$ approaches infinity

Question

If $1 > p - 1 > 0$, such as $0.5$, then $t^{p - 1}$ wouldn’t approach infinity as $t$ approaches infinity. Isn't that right? Doesn't this fact make the solution slightly off?

Answer 1

$1 > p-1 > 0$ is the same as $2 > p > 1$ for which this integral $\int t^{-p}dt$ converges. In fact it converges for all $p > 1$ as the solution states.

In response to your comment: $t^n \to \infty$ as $t\to\infty$ if $n>0$, which includes things such as $n=\frac 1 2 \text{ or } 0.1$ etc.