Consider the function defined by the limit $y(x) = \lim_{n o \infty} \frac{x}{1 + (2 \sin x)^{2n}}

Question

Consider the function defined by the limit $y(x) = \lim_{n 	o \infty} \frac{x}{1 + (2 \sin x)^{2n}}$. Which of the following statements about the continuity of $y(x)$ is/are CORRECT?

Professor PC Thomas Classes · Answer

Let $u = (2 \sin x)^2$. If $u < 1$ (i.e., $|\sin x| < 1/2$), $u^n o 0$, so $y(x) = x$. This occurs in $(-\pi/6, \pi/6)$ and periodically. Thus (B) is correct. If $u > 1$ (i.e., $|\sin x| > 1/2$), $u^n o \infty$, so $y(x) = 0$. This occurs in $(\pi/6, 5\pi/6)$ and periodically, making (C) correct. If $u = 1$, $|\sin x| = 1/2$ which implies $x = n\pi \pm \pi/6$. Here $y(x) = x/2$. Because the left-hand limit, right-hand limit, and functional values are distinct (e.g., at $\pi/6$, LHL is $\pi/6$, RHL is $0$, value is $\pi/12$), these are jump discontinuities, not removable. Thus (A) is correct and (D) is incorrect.