Limit And Continuity

Step 1: Expressing in Exponential Form Given that: Using Euler’s formula, we can rewrite it as:
Step 2: Computing the Product The product is:
Step 3: Evaluating the Summation We use the formula for the sum of the first natural numbers: Thus, the exponent simplifies to:
Step 4: Taking the Limit as Therefore:
Step 5: Verifying the Correct Option Since we found: This matches option (C) .

Step 1: Understanding the Limit We need to evaluate the limit:
Step 2: Approximate the Numerator Using the first-order approximation for small , we get: Thus, the numerator simplifies as: Since , we conclude: Thus, the numerator behaves as and requires a second-order approximation.
Step 3: Approximate the Denominator For small , Also, using , the denominator simplifies as:
Step 4: Compute the Limit Since the numerator's second-order term is: we substitute this into the fraction: Canceling , we obtain:
Step 5: Matching with the Options The calculated limit is , which matches option (D). Final Answer: The limit evaluates to (D) .