Question

The value of \(\underset{x\rightarrow 3}{lim}\frac{[x]-3}{x-3}\), where \([\cdot ]\) denotes the greatest integer function, is.....

• A. \(\infty\)
• B. \(1\)
• C. \(0\)
• D. does not exist

Hint:

Compare the left-hand and right-hand limits.

Answer 1

The Correct Option is D

Step 1: Use test values
At $x=3.001$ the value is $0$. At $x=2.999$ the value is $\frac{-1}{-0.001}=1000$, and it grows as $x$ gets closer to $3$.
Different behaviour from each side, so the limit does not exist. Option (D).

Final Answer:
Option (D). \[ \boxed{\text{(D)}} \]