Match the limit

Calculus Level 2

Let

$f(x)=\frac{a_0 x^{m}+a_1 x^{m+1}+\cdots +a_k x^{m+k}}{b_0 x^{n}+b_1 x^{n+1}+\cdots +b_ l x^{n+l}},$

where $a_0 \neq 0, b_0 \neq 0,$ and $m,n \in \mathbb N.$

Then given (A), (B), (C), or (D), $\displaystyle\lim_{x\rightarrow 0}f(x)$ equals which of (1), (2), (3), and (4)?

Match the columns:

Column-I	Column-II
(A) if $m>n$	(1) $\infty$
(B) if $m=n$	(2) $-\infty$
(C) if $m<n,$ $n-m$ is even, and $\frac{a_0}{b_0}>0$ $\hspace{10mm}$	(3) $\frac{a_0}{b_0}$
(D) if $m<n,$ $n-m$ is even, and $\frac{a_0}{b_0}<0$ $\hspace{10mm}$	(4) $0$

Note: For example, if (A) correctly matches (1), (B) with (2), (C) with (3), and (D) with (4), then answer as 1234.

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