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I just need to know whether I am right or not:If P be polynomial of degree k>0, with nonzero constant.Then $\lim_{n\rightarrow \infty}f_n(x)=P(x/n)=P(0) \forall x\in (0,\infty)$ am I right?

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    yes thank you Martin2012-06-01

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One needn't even require that the constant term be non-0 , nor restrict the value of x to only positive reals.

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Yes. If $f$ is continuous at $a$, then $\lim_{x\to a} f(g(x))=f(\lim_{x\to a}g(x))$.