Limit of a Function in a Specific Domain?

📌 CONCEPT: The limit of a function as x approaches a specific value is the value that the function approaches as x gets arbitrarily close to that value.

📐 RULE / FORMULA: According to the properties of limits, the limit of a function as x approaches a value is not affected by the function's behavior at that value.

💡 WORKED EXAMPLE: Consider the function f(x) = (2x - 4) / (x - 2). As x approaches 2, the function approaches the value 2. However, if we substitute x = 2 into the function, we get undefined, indicating that the function is not defined at x = 2. Despite this, the limit as x approaches 2 exists and is equal to 2.

⚠️ COMMON MISTAKE: Students often mistakenly assume that the limit of a function does not exist if the function is not defined at the point of interest, but this is not always the case. A limit exists if the function approaches a specific value as x gets arbitrarily close to the point, regardless of whether the function is defined at that point.