Q.$\lim_{x \to 0} \dfrac{|\sin x|}{x}$ is
(A) $1$
(B) $-1$
(C) does not exist
(D) None of these

Concept: One-sided limits and the absolute-value function.

The standard limit $\lim_{x \to 0} \frac{\sin x}{x} = 1$ holds when $x$ approaches zero from either side. Here the absolute value forces us to check left and right limits separately.

For $x \to 0^+$ (right-hand limit), $x > 0$ so $\sin x > 0$ near zero, hence $|\sin x| = \sin x$:

For $x \to 0^-$ (left-hand limit), $x < 0$ so $\sin x < 0$ near zero, hence $|\sin x| = -\sin x$: …

The absolute value introduces a directional dependence. When $x$ is near zero, $\sin x$ has the same sign as $x$ itself (positive for positive $x$, negative for negative $x$). This means $|\sin x|$ behaves differently on either side of the origin, and we must check the left-hand and right-hand limits separately.

The standard result $\lim_{x \to 0} \frac{\sin x}{x} = 1$ is fundamental, but here the absolute value changes the game.

Step-by-step analysis

1. Right-hand limit ($x \to 0^+$):

   When $x > 0$ (approaching from the right), we have $\sin x > 0$ for small positive $x$, so $|\sin x| = \sin x$.

2. Left-hand limit ($x \to 0^-$):

   When $x < 0$ (approaching from the left), we have $\sin x < 0$ for small negative $x$, so $|\sin x| = -\sin x$.

3. Comparing the one-sided limits: …