Question

The value of $\lim _{\mathrm{x} \rightarrow 0}\left[\frac{|\sin \mathrm{x}|}{|\mathrm{x}|}\right]$, (where [.] denotes greatest integer function) is (A) 0 (B) does not exists (C) \(-1\) (D) 1

Short answer

Answer: (D) 1

Step-by-step solution

Analyzing the function inside the limit

We will first analyze the function inside the limit, which is \(\frac{|\sin x|}{|x|}\). Note that it is in terms of absolute values, which means that both the numerator and denominator are always positive or non-negative for any x ≠ 0.

Determine the limit without the greatest integer function

Before applying the greatest integer function, let's find the limit as x approaches 0 of \(\frac{|\sin x|}{|x|}\). We know that \(\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1\) through L'Hopital's Rule, since we have 0/0 indeterminate form. Since \(|x|\) is the same as \(x\) when approaching 0, we also have \(\lim_{x \rightarrow 0} \frac{|\sin x|}{|x|} = 1\).

Apply greatest integer function and find the limit

Now, we are supposed to find the limit as x approaches 0 for the greatest integer function \([\frac{|\sin x|}{|x|}]\). Since we know that the limit of \(\frac{|\sin x|}{|x|}\) is 1 as x approaches 0, the value of the function inside the greatest integer function approaches 1. As the function approaches 1 (from either side), the greatest integer function will always be 1 when the input is exactly 1, and 0 when the input is slightly less than 1 (for instance, 0.9999). Therefore, the limit of the greatest integer function of this expression as x approaches 0 is 1. So, the correct answer is: (D) 1