Question

Let $f(x)=x^{6}+2 x^{4}+x^{3}+2 x+3, x \in$ R. Then the natural number $n$ for which $\displaystyle\lim _{ x \rightarrow 1} \frac{ x ^{ n } f (1)- f ( x )}{ x -1}=44$ is _____

Answer 1

$f(n)=x^{6}+2 x^{4}+x^{3}+2 x+3$
$\displaystyle\lim _{x \rightarrow 1} \frac{x^{n} f(1)-f(x)}{x-1}=44$
$\displaystyle\lim _{x \rightarrow 1} \frac{9 x^{n}-\left(x^{6}+2 x^{4}+x^{3}+2 x+3\right)}{x-1}=44$
$\displaystyle\lim _{x \rightarrow 1} \frac{9 n x^{n-1}-\left(6 x^{5}+8 x^{3}+3 x^{2}+2\right)}{1}=44$
$\Rightarrow 9 n-(19)=44$
$\Rightarrow 9 n=63$
$\Rightarrow n=7$