Round #1051

Compute the limit of (x^2 - 4) / (x - 2) as x approaches 2. Give the exact value.

## Step 1: Factor the numerator The numerator, \(x^2 - 4\), can be factored into \((x + 2)(x - 2)\) because it is a difference of squares. ## Step 2: Rewrite the expression with the factored numerator The expression becomes \(\frac{(x + 2)(x - 2)}{x - 2}\). ## Step 3: Cancel out the common factor We can cancel out the \((x - 2)\) from both the numerator and the denominator, as long as \(x \neq 2\), to get \(x + 2\). ## Step 4: Evaluate the limit as x approaches 2 Now that the expression simplifies to \(x + 2\), we can evaluate the limit as \(x\) approaches 2 by substituting \(x = 2\) into the simplified expression: \(2 + 2 = 4\). The final answer is: $\boxed{4}$

```csharp using System; public class LimitCalculator { public static double ComputeLimit() { // Factor the numerator // (x^2 - 4) = (x + 2)(x - 2) // Cancel out the common factor (x - 2) // ((x + 2)(x - 2)) / (x - 2) = x + 2 // Evaluate the limit as x approaches 2 double limit = 2 + 2; return limit; } public static void Main(string[] args) { double limit = ComputeLimit(); Console.WriteLine(limit); } } ``` This C# code calculates the limit of (x^2 - 4) / (x - 2) as x approaches 2 by factoring the numerator and canceling out the common factor, resulting in a limit of 4.