Limits

Let I =
=
=

Given,

{ (a - n) n - 1 } n = 0
(a - n ) n = 1 .

Here,
[ f ' (2) = 6 and f ' (1) = 4, given ]
Applying L'Hospital's rule,
=
= [using f ' (2) = 6 and f ' (1) = 4]

Here,
= 0 +
Appling L'Hospital's rule, we get

Let us break the given expression into two parts: Step 1: Expand the integrand using binomial expansion: Integrating term by term: Step 2: Multiply by : Step 3: Expand second term : Step 4: Add both terms: Step 5: Divide by : Step 6: Take limit as : To ensure that the limit exists and is finite, the term with must vanish: Substitute into the constant term: Oops! That's negative. But from the original equation, the limit is given as , so set: Now substitute back : Wait, the original expression gives **positive** , so we **must choose and such that the coefficient of vanishes**: Let’s re-approach: So, the final limit: