Originally Posted by

**pankaj** $\displaystyle p(x)=a_{0}+a_{1}x+a_{2}x^2+.....+a_{n}x^n$

and

$\displaystyle q(x)=b_{0}+b_{1}x+b_{2}x^2+.....+b_{m}x^m$

$\displaystyle

\lim_{x\to 0}\frac{p(x)}{q(x)}=\frac{a_{0}}{b_{0}}

$

Nothing special about this.

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Now consider

$\displaystyle p(x)=a_{0}x^n+a_{1}x^{n+1}+a_{2}x^{n+2}2+.....+a_{ n}x^{n+k}$

$\displaystyle q(x)=b_{0}x^m+b_{1}x^{m+1}+b_{2}x^{m+2}+.....+b_{n }x^{m+l}$

Evaluate $\displaystyle \lim_{x\to 0}\frac{p(x)}{q(x)}$ under the following conditions:

(i)$\displaystyle n=m$

(ii)$\displaystyle n>m,n-m$ is even

(iii)$\displaystyle n<m,n-m$ is even, $\displaystyle \frac{a_{0}}{b_{0}}>0$

(iv)$\displaystyle n<m,n-m$ is even,$\displaystyle \frac{a_{0}}{b_{0}}<0$