So while working on some calculus for distributions I came across the following integral:
$ \int_{\mathbb{R}^+} exp(\lambda x) \varphi dx $
Where $\varphi$ is a test function of compact support. To evaluate such an integral i first thought about doing an integration by parts. But while doing this I came up with the limit:
$lim_{x \rightarrow\infty} exp(\lambda x)\varphi$
Which at first glance seems to be indeterminate, is there any way to justify this limit existing or maybe another approach to make this calcuulation?