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Suppose we want to find the following limit \begin{align} \lim_{x \to \infty} \frac{f(x)}{g(x)} \end{align} and both $f(x)$ and $g(x)$ have power series represenations. Is it correct to say that \begin{align} \lim_{x \to \infty} \frac{f(x)}{g(x)}&= \lim_{x \to \infty} \frac{ \sum_1^\infty a_n x^n}{\sum_1^N b_n x^n}\\ &= \lim_{x \to \infty} \lim_{N\to \infty}\frac{ \sum_1^N a_n x^n}{\sum_1^\infty b_n x^n}\\ &= \lim_{N\to \infty}\lim_{x \to \infty} \frac{ \sum_1^N a_n x^n}{\sum_1^\infty b_n x^n}\\ &= \lim_{N \to \infty} \frac{a_N}{b_N} \end{align}

Is this justified?

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No. Counter example: $f(x)=e^x,g(x)=e^{-2x}$, then $\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=\infty$, but $f(x)=\sum\frac{x^k}{k!}$, $g(x)=\sum\frac{x^k}{k!}(-2)^k$, then $\lim_{N\rightarrow\infty}\frac{a_N}{b_N}=\lim_{N\rightarrow\infty}\frac{1}{(-2)^N}=0$.

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