- #1

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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 [itex] a [/itex] and [itex] b [/itex] are constants and [itex] g(t) [/itex] is a periodic function of [itex] t [/itex]. Now, is it correct to simply replace [itex] \lim_{x \rightarrow \infty} [/itex] by [itex] \lim_{t \rightarrow \infty} [/itex] as [itex] x \rightarrow \infty [/itex] if and only if [itex] t \rightarrow \infty [/itex]? That is, is it correct to write

\begin{equation}

\lim_{x \rightarrow \infty} f(x) = \lim_{t \rightarrow \infty} f(x(t))\;?

\end{equation}