Match the limit

Calculus Level 2

Let

f(x)=a0xm+a1xm+1++akxm+kb0xn+b1xn+1++blxn+l,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 a00,b00,a_0 \neq 0, b_0 \neq 0, and m,nN.m,n \in \mathbb N.

Then given (A), (B), (C), or (D), limx0f(x)\displaystyle\lim_{x\rightarrow 0}f(x) equals which of (1), (2), (3), and (4)?

Match the columns:

Column-IColumn-II
(A) if m>nm>n(1) \infty
(B) if m=nm=n(2) -\infty
(C) if m<n,m<n, nmn-m is even, and a0b0>0\frac{a_0}{b_0}>0 \hspace{10mm}(3) a0b0\frac{a_0}{b_0}
(D) if m<n,m<n, nmn-m is even, and a0b0<0\frac{a_0}{b_0}<0 \hspace{10mm}(4) 00

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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