I am looking for examples whose Laplace transform exists but they are not of exponential order. In the book, Schaum's Outline of Laplace Transforms, they suggest $\frac{1}{\sqrt{t}}$ is such a function. But I am not getting it.
My attempt: $\lim_{t \to \infty} \frac{e^{-\alpha t}}{\sqrt{t}} \ \to 0,\ \alpha>0$ therefore it should be of exponential order.
