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

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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.

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