MathsJEE Main 28 Jan 2026, Shift 2·Q5·MCQ·Tough

Limits, Continuity & Differentiability · Continuity of a limit-defined function

Let $f(x) = \lim_{n\to\infty}\left(\dfrac{\cos\pi x - x^{2n}\sin(x-1)}{1 + x^{2n}(x-1)}\right),\ x \in \mathbb{R}.$ Consider the following two statements :

(I) $f(x)$ is discontinuous at $x = 1$ .

(II) $f(x)$ is continuous at $x = -1$ .

Then,

(1)Neither (I) nor (II) is True
(2)Both (I) and (II) are True
(3)Only (II) is True
(4)Only (I) is True

We corrected a printing error in the original paper

Source printed “discontinous” and “continous”; source printed the limit as $\theta\to0$ with $x^{(2/\theta)}$ in place of $x^{2n}$ (undefined for $x<0$ , and the one-sided limits in $\theta$ differ for $0<\lvert x\rvert<1$ ); corrected to the intended $n\to\infty$ with $x^{2n}$ , the reading the key uses.

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See the full 28 Jan 2026, Shift 2 paper