Match the limit

Calculus Level 3

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-IColumn-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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