Limits

Step 1: Concept
Evaluate Left-Hand Limit (LHL) and Right-Hand Limit (RHL).
Step 2: Analysis
As : is a small positive decimal. Thus, for all . Numerator . RHL .
As : For odd , is a small negative decimal (e.g., -0.1). Thus, .
Step 3: Calculation
For odd (50 values), . For even (50 values), , so .
Numerator .
LHL .
Step 4: Conclusion
Since RHL ( ) LHL ( ), the limit does not exist.
Final Answer:(A)