To find the limit of the given expression as n approaches infinity, we need to simplify the expression.

3√(x³ + 3x²) / √(7x² + 2x)

First, we can divide each term in the numerator and the denominator by x³ to get rid of the highest power of x in the expression.

(3√(x³ + 3x²) / x³) / (√(7x² + 2x) / x³)

Now, we simplify the expression further by factoring out x from the square roots:

(3√(x³(1 + 3/x)) / x³) / (√(x²(7 + 2/x)) / x³)
(3x√(1 + 3/x) / x³) / (x√(7 + 2/x) / x³)

Next, we can simplify the expression by canceling out the x’s:

3√(1 + 3/x) / √(7 + 2/x)

As n approaches infinity, the terms with 3/x and 2/x approach zero. Therefore, in the limit as n approaches infinity, we are left with:

3√1 / √7 = 3 / √7 = 3√7 / 7.

So, lim n->∞ ³√(x³+3x²)/√(7x²+2x) = 3√7 / 7.