QJEEM21Ma67ac: JEE Main 2021 Mathematics - Class 12 JEEM21Ma67ac Solution: Let f : R → R be defined as f ( x ) = { x 3 ( 1 − cos ⁡ 2 x ) 2 …

Question

Verified Solution • 2025–26 Session moderate

Let f : R

\to

R be defined as

f (x) = {\begin{matrix} \frac{x^{3}}{{(1 - \cos 2 x)}^{2}} \log_{e} (\frac{1 + 2 x e^{- 2 x}}{{(1 - x e^{- x})}^{2}}), & x \neq 0 \\ α, & x = 0 \end{matrix}

If f is continuous at x = 0, then

α

is equal to :

Accepted Answer

For continuity

lim_{x \to 0} \frac{x^{3}}{4 \sin^{4} x} (\ln (1 + 2 x e^{- 2 x}) - 2 \ln (1 - x e^{- x})) = α

lim_{x \to 0} \frac{1}{4 x} [2 x e^{- 2 x} + 2 x e^{- x}] = α

= \frac{1}{4} (4) = α = 1

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