Let $f(x) = {x^6} + 2{x^4} + {x^3} + 2x + 3$, x $\in$ R. Then the natural number n for which $\mathop {\lim }\limits_{x \to 1} {{{x^n}f(1) - f(x)} \over {x - 1}} = 44$ is __________.
Answer (integer)
7
Solution
$f(x) = {x^6} + 2{x^4} + {x^3} + 2x + 3$<br><br>$\mathop {\lim }\limits_{x \to 1} {{{x^n}f(1) - f(x)} \over {x - 1}} = 44$<br><br>$$\mathop {\lim }\limits_{x \to 1} {{9{x^n} - ({x^6} + 2{x^4} + {x^3} + 2x + 3)} \over {x - 1}} = 44$$<br><br>$$\mathop {\lim }\limits_{x \to 1} {{9n{x^{n - 1}} - (6{x^5} + 8{x^3} + 3{x^2} + 2)} \over 1} = 44$$<br><br>$\Rightarrow$ 9n $-$ (19) = 44<br><br>$\Rightarrow$ 9n = 63<br><br>$\Rightarrow$ n = 7
About this question
Subject: Mathematics · Chapter: Limits, Continuity and Differentiability · Topic: Limits and Standard Results
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