Limits

Step 1: Analyze the First Term
First, simplify the term .
For small , we know: Thus: Now divide by : As , this term approaches 0.
Step 2: Analyze the Second Term
The term oscillates between -1 and 1 for all , but does not affect the limit since it is bounded.
Step 3: Conclusion
Since the first term approaches 0, the whole product approaches 0.
Hence, the limit is 0.

Step 1: Simplify the Expression
To compute this limit, use the fact that for small , . Thus: Substituting these approximations into the given expression:
Step 2: Evaluating the Limit
Taking the limit as , the constant terms cancel out, and we are left with:
Step 3: Conclusion
Thus, the limit is equal to 1.