Limits, Continuity & Differentiability
Limits with Greatest Integer Function
Grade 12
<p>The value of $\lim_{x \to 0} \frac{\sin[x]}{[x]}$ (where $[\cdot]$ denotes the greatest integer function) is</p>
Step-by-Step Solution
Key Concept: The greatest integer function approaches different values from left and right near 0, causing the limit to fail to exist.
<p>As $x \to 0^+$: $[x] = 0$, so the expression becomes $\frac{\sin 0}{0}$ which is indeterminate.</p><p>As $x \to 0^-$: $[x] = -1$, so the expression becomes $\frac{\sin(-1)}{-1} = \frac{-\sin 1}{-1} = \sin 1$</p><p>Since the left and right hand limits are different (or one side is indeterminate), the limit doesn't exist.</p>
Correct Answer: C