QJEEM21Maa91e: JEE Main 2021 Mathematics - Class 12 JEEM21Maa91e Solution: Let f : R R be a function such that f(2) = 4 and f'(2) = 1.…

This limit can be solved using L'Hopital's Rule, which states that for the limit of the form 0/0 or ±∞/±∞, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately.

$\mathop {\lim }\limits_{x \to 2} {{{x^2}f(2) - 4f(x)} \over {x - 2}}$ is in the indeterminate form, and we are given that f(2) = 4 and f'(2) = 1, so we can apply L'Hopital's rule.

Taking the derivative of the numerator and the denominator, we get :

Numerator: derivative of $x^2 f(2) - 4f(x)$ is $2x f(2) - 4f'(x)$ .

Denominator: derivative of $x - 2$ is $1$ .

So, the limit becomes :

$\mathop {\lim }\limits_{x \to 2} {{{2xf(2) - 4f'(x)}} \over 1} = 2 \times 2 \times f(2) - 4 \times f'(2) = 16 - 4 = 12.$

Therefore, Option D, 12, is the correct answer.