Limit Of (2x^2 - 8)/(x - 4) As X Approaches 4

Let's dive into finding the limit of the function f(x) = (2x^2 - 8)/(x - 4) as x approaches 4. This is a classic calculus problem where we need to be careful about direct substitution because it might lead to an indeterminate form. To solve this, we'll use algebraic manipulation to simplify the function before evaluating the limit. Understanding limits is super important because it forms the foundation for calculus concepts like derivatives and integrals. So, let's break it down step by step to make sure we get it right. When we're dealing with limits, the first thing we often try is to directly substitute the value that x is approaching into the function. However, if we do that here, we quickly run into a problem. Specifically, if we plug in x = 4 into the function, we get f(4) = (2*(4^2) - 8)/(4 - 4) = (2*16 - 8)/0 = (32 - 8)/0 = 24/0. Uh oh! We can't divide by zero; it's undefined. This tells us that we can't just directly substitute x = 4 to find the limit. Instead, we need to do some algebraic gymnastics to see if we can simplify the expression. The goal is to manipulate the function so that we can get rid of the problematic term in the denominator that's causing the division by zero. One common technique is to factor the numerator and see if any terms cancel out with the denominator. Let's give that a try. Looking at the numerator, 2x^2 - 8, we can factor out a 2: 2x^2 - 8 = 2(x^2 - 4). Now, we recognize that x^2 - 4 is a difference of squares, which can be further factored as (x - 2)(x + 2). So, the numerator becomes 2(x - 2)(x + 2). Putting it all together, our function now looks like this: f(x) = (2(x - 2)(x + 2))/(x - 4). Unfortunately, we don't see any immediate cancellations between the numerator and the denominator. The denominator is (x - 4), and we don't have that term in the numerator. This means we need to rethink our approach. We go back to the original function: f(x) = (2x^2 - 8)/(x - 4). Factoring out 2 in the numerator gives us f(x) = 2(x^2 - 4)/(x - 4). Now, recognize x^2 - 4 as a difference of squares, so x^2 - 4 = (x - 2)(x + 2). Therefore, we can rewrite the function as f(x) = 2(x - 2)(x + 2) / (x - 4). Still, there are no common factors to cancel out. This suggests there might be an error, or perhaps the limit does not exist in the way we initially hoped. However, let's try a different factorization. Notice that we want to evaluate the limit as x approaches 4, so let's rewrite the numerator in terms of (x - 4). We can express 2x^2 - 8 as 2(x^2 - 4), which factors into 2(x - 2)(x + 2). This doesn't directly help us cancel out the (x - 4) term in the denominator. Let's go back and check our initial algebra to make sure we didn't miss anything. We have f(x) = (2x^2 - 8)/(x - 4). If we substitute x = 4, we get (2(4)^2 - 8)/(4 - 4) = (32 - 8)/0 = 24/0, which is undefined. This confirms that direct substitution won't work. Let’s rethink the factorization. We have f(x) = (2x^2 - 8) / (x - 4). We can factor a 2 out of the numerator to get f(x) = 2(x^2 - 4) / (x - 4). Now, we recognize that x^2 - 4 is a difference of squares, so we can factor it as (x - 2)(x + 2). Thus, f(x) = 2(x - 2)(x + 2) / (x - 4). Still no direct cancellation! Hmmm. Let's try polynomial long division to divide 2x^2 - 8 by x - 4. Before we do that, it's worth noting that if we can somehow manipulate the numerator to have a factor of (x - 4), then we can cancel it with the denominator. But as it stands, it doesn't look like a straightforward factorization will get us there. Polynomial long division might give us some insight. If we perform polynomial long division of (2x^2 - 8) by (x - 4), we get: 2x + 8 with a remainder of 24. This means that 2x^2 - 8 = (x - 4)(2x + 8) + 24. So, f(x) = ((x - 4)(2x + 8) + 24) / (x - 4) = (2x + 8) + 24/(x - 4). Now, we take the limit as x approaches 4: lim (x→4) [(2x + 8) + 24/(x - 4)]. As x approaches 4, 2x + 8 approaches 2(4) + 8 = 16. However, the term 24/(x - 4) approaches infinity as x approaches 4 (from either the left or the right). Thus, the limit does not exist as a finite number. If x approaches 4 from the right (x > 4), then x - 4 is a small positive number, and 24/(x - 4) approaches +∞. If x approaches 4 from the left (x < 4), then x - 4 is a small negative number, and 24/(x - 4) approaches -∞. Since the limit from the right and the limit from the left are not the same, the limit does not exist. Given the alternatives, the closest answer would be [infinity], but it's important to note that the limit doesn't technically exist because it goes to infinity. This is a good reminder that sometimes limits can be tricky and require a mix of algebraic techniques and careful analysis. So, to recap, we started with the function f(x) = (2x^2 - 8)/(x - 4) and wanted to find the limit as x approaches 4. We tried direct substitution and found it to be undefined (division by zero). We then attempted factoring but couldn't find a way to cancel out the (x - 4) term in the denominator. Finally, we performed polynomial long division and found that the function could be rewritten as (2x + 8) + 24/(x - 4). Analyzing this form, we saw that as x approaches 4, the term 24/(x - 4) goes to infinity. Therefore, the limit does not exist as a finite number, and the closest answer among the alternatives is [infinity]. Remember, understanding these steps and techniques is key to mastering limits in calculus!

Understanding Limits: A Deep Dive

Let's explore the concept of limits in more detail. Limits are fundamental to calculus and are used to define continuity, derivatives, and integrals. In simpler terms, a limit tells us what value a function approaches as its input gets closer and closer to a specific value. In the context of our problem, we're looking at what happens to the function f(x) = (2x^2 - 8)/(x - 4) as x gets closer to 4. As we saw earlier, directly substituting x = 4 into the function results in an undefined expression because we end up dividing by zero. This is a common situation when dealing with limits, and it often requires us to manipulate the function algebraically to find the limit. One of the key techniques we used was factoring. Factoring allows us to simplify complex expressions and potentially cancel out terms that are causing the division by zero. In our case, we factored the numerator to see if we could find a common factor with the denominator. While we didn't find a direct cancellation, the process of factoring helped us understand the structure of the function better. Another powerful technique for evaluating limits is L'Hôpital's Rule. L'Hôpital's Rule states that if we have a limit of the form 0/0 or ∞/∞, we can take the derivative of the numerator and the derivative of the denominator and then evaluate the limit again. In our case, if we apply L'Hôpital's Rule to the original function f(x) = (2x^2 - 8)/(x - 4), we would first check that the limit is of the form 0/0 when x = 4. Indeed, as x approaches 4, the numerator 2x^2 - 8 approaches 2(4^2) - 8 = 32 - 8 = 24, and the denominator x - 4 approaches 4 - 4 = 0. So, we have a non-zero number divided by zero, which means L'Hôpital's Rule isn't directly applicable. However, let's consider a slightly modified function where direct substitution gives us 0/0. Suppose we had f(x) = (2x^2 - 32)/(x - 4). As x approaches 4, the numerator approaches 2(4^2) - 32 = 32 - 32 = 0, and the denominator approaches 4 - 4 = 0. Now we have the indeterminate form 0/0, and we can apply L'Hôpital's Rule. We take the derivative of the numerator and the derivative of the denominator: The derivative of 2x^2 - 32 is 4x. The derivative of x - 4 is 1. So, the new limit is lim (x→4) (4x/1) = 4(4) = 16. Thus, for this modified function, the limit as x approaches 4 is 16. Going back to our original function, f(x) = (2x^2 - 8)/(x - 4), we can also use the technique of rationalizing the denominator. However, in this case, it's not applicable because the denominator is already a simple expression. The key to solving this problem was understanding that the limit does not exist as a finite number. Instead, the function approaches infinity as x approaches 4. This is because the denominator gets closer and closer to zero, while the numerator approaches a non-zero value. When we divide a non-zero number by a very small number, we get a very large number, which means the function approaches infinity. It's also important to consider the limits from the left and the right. If the limit from the left is different from the limit from the right, then the overall limit does not exist. In our case, as x approaches 4 from the right (x > 4), the denominator is positive, and the function approaches positive infinity. As x approaches 4 from the left (x < 4), the denominator is negative, and the function approaches negative infinity. Since the limits from the left and the right are different, the overall limit does not exist. In summary, understanding limits involves a combination of algebraic manipulation, calculus techniques, and careful analysis. It's a fundamental concept that underlies much of calculus and is essential for solving a wide range of problems. Remember, the key is to look for ways to simplify the function and to consider the behavior of the function as the input approaches a specific value. Practice makes perfect, so keep working on these types of problems to build your skills and intuition!

Practical Applications and Real-World Examples

Limits, although abstract, have practical applications in various fields. They are essential in engineering, physics, computer science, and economics. Understanding how functions behave as they approach certain values is crucial for modeling real-world phenomena. In engineering, limits are used to analyze the stability of structures. For example, engineers need to know how a bridge will behave under increasing loads. By using limits, they can determine the point at which the bridge will become unstable and potentially collapse. This involves analyzing the stresses and strains on the bridge as the load approaches a critical value. In physics, limits are used to describe the behavior of objects at extreme conditions. For example, when studying the motion of an object near the speed of light, physicists use limits to understand how the object's mass and energy change as its velocity approaches the speed of light. This is essential for understanding Einstein's theory of relativity. In computer science, limits are used in the analysis of algorithms. Computer scientists often need to determine how the running time of an algorithm scales as the input size increases. By using limits, they can analyze the asymptotic behavior of the algorithm and determine its efficiency. This is crucial for designing algorithms that can handle large datasets. In economics, limits are used to model the behavior of markets. Economists often need to understand how the price of a product changes as the supply or demand changes. By using limits, they can analyze the equilibrium price and quantity and predict how the market will respond to changes in economic conditions. For instance, consider a scenario where a company is trying to optimize its production process. The company wants to minimize its costs while maximizing its output. By using limits, they can analyze the cost function and the production function to find the optimal production level. This involves finding the point at which the marginal cost equals the marginal revenue, which can be determined using limits. Another example is in the field of finance. Financial analysts use limits to analyze the behavior of stock prices. They often need to understand how the price of a stock will change as the trading volume increases. By using limits, they can analyze the market dynamics and predict the future behavior of the stock. This is essential for making informed investment decisions. In each of these examples, limits provide a powerful tool for understanding and predicting the behavior of complex systems. By understanding how functions behave as they approach certain values, we can gain valuable insights into the real world. So, while limits may seem abstract, they are essential for solving a wide range of practical problems. Remember, the key is to understand the underlying concepts and to apply them creatively to solve real-world problems. With practice, you can become proficient in using limits to analyze and understand the world around you. Furthermore, consider the design of aircraft wings. Engineers use limits to optimize the shape of the wings to maximize lift and minimize drag. They analyze the airflow around the wings as the speed of the aircraft approaches its cruising speed. By using limits, they can determine the optimal wing shape that will provide the best performance. This involves analyzing the pressure distribution on the wings and minimizing the turbulence. These examples illustrate the broad applicability of limits in various fields. Whether you are an engineer, a physicist, a computer scientist, or an economist, understanding limits is essential for solving complex problems and making informed decisions. So, keep practicing and keep exploring the many ways that limits can be used to understand the world around you.