Question:
Let $f(x)=x^{6}+2 x^{4}+x^{3}+2 x+3, x \in \mathbf{R}$. Then the natural number $n$ for which $\lim _{x \rightarrow 1} \frac{x^{n} f(1)-f(x)}{x-1}=44$ is
Solution:
$f(n)=x^{6}+2 x^{4}+x^{3}+2 x+3$
$\lim _{x \rightarrow 1} \frac{x^{n} f(1)-f(x)}{x-1}=44$
$\lim _{x \rightarrow 1} \frac{9 x^{n}-\left(x^{6}+2 x^{4}+x^{3}+2 x+3\right)}{x-1}=44$
$\lim _{x \rightarrow 1} \frac{9 n x^{n-1}-\left(6 x^{5}+8 x^{3}+3 x^{2}+2\right)}{1}=44$
$\Rightarrow 9 \mathrm{n}-(19)=44$
$\Rightarrow 9 \mathrm{n}=63$
$\Rightarrow \mathrm{n}=7$
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