Applications Of Integrals

To find the value of the given limit, we need to analyze the expression:

We will solve this using the properties of limits and exponentials:

1. Observe that approaches 1 as goes to infinity for each .
2. Thus, the sum in the numerator becomes:
3. Therefore, the overall expression inside the limit simplifies to:
4. Now we focus on the term raised to the power of :
5. However, using the exponential limit property, if we rewrite the exponent:
   For large ,
6. Substituting this, the original expression becomes:
7. Simplified as:
8. Using the limit property, as , we get:

Therefore, the value of the limit is:

This matches with the given correct answer option: