@Useful nucleus and alexfloo: Thanks!
You are are right, "a" has to be a positive real number and yes the only requirement on g(t) should be as you have stated.
This is not the same [type of] question I have asked.
I should have stated this more explicitly: \left|g(t)\right| < \infty , say \left|g(t)\right| = \sin(t) , continuous, smooth and infinitely differentiable.
I have a rather silly limit question.
Consider
\begin{equation}
\lim_{x \rightarrow \infty} f(x)
\end{equation}
and assume it exists. Suppose now that
\begin{equation}
x = a\, t + b\, g(t),
\end{equation}
where a and b are constants and g(t) is a periodic function of t . Now...