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I came across a problem which says:

Let $P$ be a polynomial of degree $k>0$ with a non-zero constant term.Let $f_n(x)=P(x/n), \, \forall x \in (0,\infty)$. Then which of the following is/are true?

(a) $\lim f_n(x)=\infty \quad \forall x \in (0,\infty)$

(b) $\exists x \in (0,\infty)$ such that $\lim f_n(x)>P(0)$

(c) $\lim f_n(x)=0 \quad \forall x \in (0,\infty)$

(d) $\lim f_n(x)=P(0) \quad \forall x \in (0,\infty)$

I do not know how to approach the problem. Any kind of hints will be helpful. Thanks in advance for your time.

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Please do use LaTeX to write mathematics in an appropriate way. You can find directions in teh FAQ section. – DonAntonio Nov 20 '12 at 18:43
And please fix the title yourself. I tried to turn it into LaTeX, but then it was too long, and it should probably be shortened. – Lukas Geyer Nov 20 '12 at 18:48
up vote 3 down vote accepted

Hints: 1) Let the polynomial be $\sum_{j=0}^k a_j x^j$.

Consider the term $a_jx^j$. Let $j\gt 0$, and let $x$ be a fixed number. How does $a_j\left(\dfrac{x}{n}\right)^j$ behave as $n$ gets large?

2) Since this is multiple choice, test the proposed "facts" against simple polynomials.

For example, for (a), think of the polynomial $P(x)=x+1$. Does $\frac{x}{n}+1$ get large as $n$ gets large? The same polynomial will let you settle (b) and (c) in the negative. And it will get you close to the answer for (d).

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Every polynomial is a continuous function in $\mathbb{R}$.
Since $\displaystyle{\lim_{n \to \infty} \frac{x}{n}=0}$ we obtain $\displaystyle{\lim_{n \to \infty}P\left(\frac{x}{n}\right)=P(0)} \ \forall x \in \mathbb{R}$.

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