Limits And Derivatives

Let .....(i)
Taking log in (i) on both sides, we get

Applying L'-Hospital's Rule, we get

or

We are given the limit: Step 1: Simplify the expression Factor out from the numerator: Simplifying: Step 2: Evaluate the limit As , the expression simplifies to: Since , the limit does not exist, so the correct answer is (d) None of these.

We are given the limit: Step 1: Apply binomial expansion We first expand using the binomial expansion for small : For small , we approximate this expansion as: Step 2: Simplify the limit Substitute this approximation into the limit expression: Simplify the numerator: For to be finite as , we must have . Thus, the correct answer is .