Question

Find the limit: \[ \lim_{x \to 0} \frac{\sin[x]}{[x]}, \text{ where } [x] \text{ represents greatest integer function} \]

• A. 0
• B. 1
• C. Does not exist
• D. Infinity

Hint:

When dealing with the greatest integer function, be aware of discontinuities where the function "jumps" at integer values.

Answer 1

The Correct Option is A

The greatest integer function \( [x] \) gives the greatest integer less than or equal to \( x \). As \( x \to 0 \), the value of \( [x] \) approaches 0. The sine function is continuous at 0, and \( \sin(0) = 0 \). Therefore, the limit becomes: \[ \lim_{x \to 0} \frac{\sin[x]}{[x]} = \frac{0}{0} = 0 \] Thus, the limit is 0.