I would like to know if there is any problem with defining the following expression: $$ I = \int_0^\infty g(t) \delta(f(t))\mathrm{d}t $$ where $0<\lim\limits_{t\to\infty} g(t) =L<\infty$ and $f(t)>0$ for all $t\in[0,\infty)$ but $\lim\limits_{t\to\infty} f(t) =0$, such that
$$I = \lim\limits_{t\to\infty} g(t) = L$$ For example, is the following correct? $$ \int_0^\infty \underbrace{\sqrt{1+\frac{1}{t}}}_{=g(t)} \text{ }\delta\left(\frac{1}{t}\right)\mathrm{d}t = \lim\limits_{t\to\infty} g(t) = 1 $$
I was wondering if this is well-defined because one usually uses compactly supported test functions, but I do not have a thorough training on distributions yet.