I stumbled accross this question and would like to know if my approach makes sense, and if there is a better one:
For which values of $t$ does the following integral converge?
$$\int_1^\infty e^{1/x}-t(\sin(\frac1x)+\cos(\frac1x)) dx$$
I thought of using Taylor's polynomial to the 2nd degree, and using Lagrange's form of the remainder block the integral between two easier integrals, then showing it converges iff $t=1$. Is there a better approach?