## Intuition The problem requires us to find the limit of the ratio of two polynomials as \( x \) approaches infinity. The behavior of the limit is determined by the degrees of the polynomials and their leading coefficients. Specifically: 1. If the degree of \( P(x) \) (numerator) is greater than the degree of \( Q(x) \) (denominator), the limit is \( \pm \infty \) depending on the sign of the leading coefficients. 2. If the degree of \( P(x) \) is less than the degree of \( Q(x) \), the limit is 0. 3. If the degrees are equal, the limit is the ratio of the leading coefficients. ## Approach 1. **Read Input:** Parse the degrees and coefficients of both polynomials. 2. **Compare Degrees:** - If \( n > m \), the limit is \( \pm \infty \). Determine the sign by comparing the leading coefficients. - If \( n < m \), the limit is 0. - If \( n = m \), compute the limit as the ratio of the leading coefficients. 3. **Output Result:** Depending on the comparison, output "Infinity", "-Infinity", "0/1", or the irreducible fraction of the leading coefficients. ## Complexity - **Time:** \( O(1) \) - The solution involves a constant amount of work regardless of the size of the input. - **Space:** \( O(1) \) - We use a fixed amount of extra space. ## Solution (C++) ```cpp #include using namespace std; int main() { ios::sync_with_stdio(false); cin.tie(nullptr); int n, m; cin >> n >> m; vector a(n + 1), b(m + 1); for (int i = 0; i <= n; ++i) { cin >> a[i]; } for (int i = 0; i <= m; ++i) { cin >> b[i]; } if (n > m) { if ((a[0] > 0 && b[0] > 0) || (a[0] < 0 && b[0] < 0)) { cout << "Infinity" << endl; } else { cout << "-Infinity" << endl; } } else if (n < m) { cout << "0/1" << endl; } else { int gcd = __gcd(abs(a[0]), abs(b[0])); int p = a[0] / gcd; int q = b[0] / gcd; if (q < 0) { p = -p; q = -q; } cout << p << "/" << q << endl; } return 0; } ``` This solution directly implements the logic derived from the polynomial degree comparison and outputs the correct limit based on the conditions specified.

Limit

Problem Details