Round #1097

Compute the limit of the sequence a_n = (5n^2 + 2) / (2n^2 - n + 4) as n approaches infinity. Give the exact value.

## Step 1: Identify the sequence and the limit in question The sequence given is \(a_n = \frac{5n^2 + 2}{2n^2 - n + 4}\), and we need to find the limit as \(n\) approaches infinity. ## Step 2: Determine the method to evaluate the limit To evaluate the limit of a sequence as \(n\) approaches infinity, especially for rational functions like this one, we can compare the degrees of the polynomials in the numerator and denominator. If they are the same, the limit will be the ratio of the leading coefficients. ## Step 3: Compare the degrees of the numerator and denominator The degree of the numerator \(5n^2 + 2\) is 2, and the degree of the denominator \(2n^2 - n + 4\) is also 2. Since they have the same degree, the limit as \(n\) approaches infinity will be the ratio of the leading coefficients. ## Step 4: Calculate the limit The leading coefficient of the numerator is 5, and the leading coefficient of the denominator is 2. Therefore, the limit as \(n\) approaches infinity is \(\frac{5}{2}\). The final answer is: $\boxed{\frac{5}{2}}$

To compute the limit of the sequence `a_n = (5n^2 + 2) / (2n^2 - n + 4)` as `n` approaches infinity, we can use the following C# code: ```csharp using System; public class LimitCalculator { public static double CalculateLimit() { // Define the function for the sequence Func<double, double> sequence = n => (5 * Math.Pow(n, 2) + 2) / (2 * Math.Pow(n, 2) - n + 4); // Calculate the limit as n approaches infinity double limit = 5.0 / 2.0; // Since the highest order terms dominate as n approaches infinity return limit; } public static void Main(string[] args) { double limit = CalculateLimit(); Console.WriteLine("The limit of the sequence as n approaches infinity is: " + limit); } } ``` This code calculates the limit by dividing the highest order terms in the numerator and denominator, which dominate as `n` approaches infinity. The result is `5/2` or `2.5`. Alternatively, we can apply L'Hôpital's rule, but in this case, it's not necessary since the limit can be evaluated directly by considering the dominant terms. The limit of the sequence `a_n` as `n` approaches infinity is `2.5`.