QJEEM21Ma7810: JEE Main 2021 Mathematics - Class 12 JEEM21Ma7810 Solution: Let f ( x ) = x 6 + 2 x 4 + x 3 + 2 x + 3 , x ∈ R. Then the…

Question

Verified Solution • 2025–26 Session moderate

Let

f (x) = x^{6} + 2 x^{4} + x^{3} + 2 x + 3

, x

\in

R. Then the natural number n for which

lim_{x \to 1} \frac{x^{n} f (1) - f (x)}{x - 1} = 44

is __________.

Accepted Answer

f (x) = x^{6} + 2 x^{4} + x^{3} + 2 x + 3

lim_{x \to 1} \frac{x^{n} f (1) - f (x)}{x - 1} = 44

lim_{x \to 1} \frac{9 x^{n} - (x^{6} + 2 x^{4} + x^{3} + 2 x + 3)}{x - 1} = 44

lim_{x \to 1} \frac{9 n x^{n - 1} - (6 x^{5} + 8 x^{3} + 3 x^{2} + 2)}{1} = 44

\Rightarrow

9n

-

(19) = 44

\Rightarrow

9n = 63

\Rightarrow

n = 7

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