Find the positive integer “n” so that limx → 3[(xn– 3n)/(x – 3)] = 108.

As an experienced tutor registered on UrbanPro, I can confidently guide you through this problem using my expertise. UrbanPro is indeed a fantastic platform for online coaching and tuition, offering a diverse range of subjects and experienced tutors like myself. Now, let's tackle the problem at hand. We're given the expression lim(x → 3) [(x^n – 3^n) / (x – 3)] = 108, where we need to find the positive integer "n". First, let's notice that the expression looks very similar to the definition of the derivative of a function at a point. It resembles the difference quotient: f'(3) = lim(x → 3) [(f(x) - f(3)) / (x - 3)] Given that our expression equals 108, it seems like we're dealing with a derivative. To find "n", we need to find a function f(x) whose derivative at x = 3 is equal to 108. Let's try to identify such a function. Since the derivative of a constant is zero, we can't have a constant function. We need a function whose rate of change at x = 3 equals 108. One such function is f(x) = x^n. Its derivative with respect to x is n*x^(n-1). At x = 3, this derivative would be n * 3^(n-1). So, we need to find "n" such that n * 3^(n-1) = 108. Let's try some values of n: For n = 4, we get 4 * 3^(4-1) = 4 * 3^3 = 4 * 27 = 108. This satisfies the equation. Therefore, the positive integer "n" that satisfies lim(x → 3) [(x^n – 3^n) / (x – 3)] = 108 is n = 4. UrbanPro provides a platform for students to connect with knowledgeable tutors who can break down complex problems like this one. If you have any further questions or need additional clarification, feel free to ask! read less