$f(x)=x^6+2x^4+x^3+2x+3 $$\mathop {\lim }\limits_{x \to 1} \frac{{{x^n}f(1) - f(x)}}{{x - 1}} = 44$$n=?$

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

Solution

Since the limit has $\left[ {\frac{0}{0}} \right]$ form, L.H. Rule is applicable.

Thus, $\mathop {\lim }\limits_{x \to 1} n{x^{n - 1}}f(1) - f'(x) = 44$

$\therefore nf(1) - f'(1) = 44$

$\therefore n.9 - ({6.1^5} + {8.1^3} + {3.1^2} + 2.1) = 44$

$ \Rightarrow 9n - 19 = 44$

$\Rightarrow n=7$