Round #1081

Compute the limit of (x^2 - 9) / (x - 3) as x approaches 3. Give the exact value.

## Step 1: Factor the numerator The numerator, \(x^2 - 9\), can be factored into \((x + 3)(x - 3)\) because it follows the difference of squares formula, \(a^2 - b^2 = (a + b)(a - b)\), where \(a = x\) and \(b = 3\). ## Step 2: Rewrite the expression with the factored numerator Substitute the factored form of the numerator back into the original expression to get \(\frac{(x + 3)(x - 3)}{x - 3}\). ## Step 3: Cancel out the common factor Notice that \((x - 3)\) appears in both the numerator and the denominator. As long as \(x \neq 3\), we can cancel out these terms, leaving us with \(x + 3\). ## Step 4: Evaluate the limit Since we are asked to find the limit as \(x\) approaches 3, we substitute \(x = 3\) into the simplified expression \(x + 3\). Thus, \(3 + 3 = 6\). The final answer is: $\boxed{6}$

```csharp using System; public class LimitCalculator { public static double ComputeLimit() { // Factor the numerator // (x^2 - 9) = (x - 3)(x + 3) // So, the expression becomes ((x - 3)(x + 3)) / (x - 3) // Cancel out the common factor (x - 3) // The expression simplifies to (x + 3) // Now, we can directly substitute x = 3 into the simplified expression double result = 3 + 3; return result; } public static void Main(string[] args) { double limitValue = ComputeLimit(); Console.WriteLine(limitValue); } } ``` This C# code calculates the limit of the given expression as x approaches 3 by factoring the numerator, canceling out the common factor, and then directly substituting x = 3 into the simplified expression. The result is the exact value of the limit.