Find The Limit Of The Function (5+√x)/√(x+36) As X Approaches 0 From The Right, Rounded To The Nearest Hundredth.

In the realm of calculus, exploring the behavior of functions as their input approaches a specific value is a fundamental concept. Limits, in particular, provide a powerful tool for understanding how a function behaves near a point, even if the function is not defined at that exact point. In this article, we will delve into the process of finding the limit of the function f(x) = (5+√x)/√(x+36) as x approaches 0 from the right-hand side, denoted as x → 0⁺. This exploration will involve a careful examination of the function's components and how they interact as x gets infinitesimally close to 0, but remains strictly greater than 0. We'll employ a combination of algebraic manipulation and limit laws to arrive at the solution, and then round the result to the nearest hundredth, providing a practical and readily interpretable answer. Understanding limits is crucial not only for grasping the theoretical underpinnings of calculus but also for applying these concepts to real-world problems in physics, engineering, and other scientific disciplines. This journey into the world of limits will showcase the elegance and precision of mathematical analysis, enabling us to predict and understand the behavior of functions in a rigorous and insightful manner.

Understanding Limits and Their Significance

Before diving into the specifics of our problem, let's take a moment to appreciate the broader concept of limits in calculus. A limit, informally speaking, is the value that a function "approaches" as the input (or argument) approaches a certain value. It's important to emphasize that the limit doesn't necessarily describe the actual value of the function at that point, but rather the value it gets arbitrarily close to. This subtle distinction is crucial, especially when dealing with functions that are undefined at a particular point or exhibit discontinuous behavior. The notation limₓ→ₐ f(x) = L signifies that as x gets closer and closer to 'a', the function f(x) approaches the value L. Limits form the bedrock of calculus, serving as the foundation for concepts like continuity, derivatives, and integrals. They allow us to analyze the behavior of functions in a nuanced way, capturing trends and tendencies that would be obscured by simply evaluating the function at a single point. For instance, consider a function that represents the velocity of an object over time. The limit of this function as time approaches a certain value would tell us the instantaneous velocity of the object at that moment, even if the velocity is changing rapidly. In this particular problem, we are interested in the limit as x approaches 0 from the right (x → 0⁺). This means we are only considering values of x that are greater than 0, but getting progressively closer to 0. This one-sided limit is relevant when the function's behavior might be different depending on whether we approach 0 from the positive or negative side.

H2: Problem Statement: Finding the Limit of the Given Function

Our primary goal is to determine the limit of the function f(x) = (5+√x)/√(x+36) as x approaches 0 from the right (x → 0⁺). This means we need to investigate the behavior of this function as x takes on values that are positive and progressively closer to 0, such as 0.1, 0.01, 0.001, and so on. To tackle this problem effectively, we will employ a combination of direct substitution (if applicable) and limit laws. Direct substitution involves simply plugging in the value that x is approaching (in this case, 0) into the function and evaluating the result. However, this method only works if the function is continuous at that point and doesn't lead to any indeterminate forms like 0/0 or ∞/∞. If direct substitution fails, we need to resort to other techniques, such as algebraic manipulation or the application of specific limit laws. Limit laws are a set of rules that allow us to break down complex limits into simpler ones. For instance, the limit of a sum is the sum of the limits, the limit of a product is the product of the limits, and so on. These laws provide a systematic way to handle various types of limit problems. In this specific case, we will first attempt direct substitution to see if it yields a determinate result. If not, we will carefully analyze the structure of the function and apply appropriate limit laws to simplify the expression and ultimately find the limit. The process will involve close attention to detail and a clear understanding of the properties of square roots and rational functions.

H2: Step-by-Step Solution: Evaluating the Limit

Let's embark on the journey of finding the limit of f(x) = (5+√x)/√(x+36) as x approaches 0 from the right. Our first step is to attempt direct substitution. This involves plugging in x = 0 into the function and observing the result. When we substitute x = 0 into the function, we get: f(0) = (5 + √0) / √(0 + 36) = (5 + 0) / √36 = 5 / 6. Since we obtained a determinate value (5/6) without encountering any indeterminate forms (such as 0/0 or ∞/∞), direct substitution is indeed a valid approach in this case. This suggests that the function is continuous at x = 0, which means that the limit as x approaches 0 is simply equal to the value of the function at x = 0. Therefore, the limit of the function as x approaches 0 from the right is 5/6. However, the problem asks us to round the solution to the nearest hundredth. To do this, we need to convert the fraction 5/6 into a decimal representation and then round it appropriately. Dividing 5 by 6, we get approximately 0.83333... The digit in the hundredths place is 3, and the digit to its right is also 3. Since the digit to the right is less than 5, we round down, keeping the hundredths digit as 3. Therefore, the limit rounded to the nearest hundredth is 0.83. It's worth noting that the success of direct substitution in this case is due to the fact that both the numerator (5 + √x) and the denominator √(x + 36) are continuous functions at x = 0, and the denominator is non-zero at that point. This allows us to evaluate the limit by simply evaluating the function at the point of interest.

H2: Conclusion: The Limiting Value and Its Significance

In conclusion, by carefully applying the concept of limits and employing direct substitution, we have successfully determined that the limit of the function f(x) = (5+√x)/√(x+36) as x approaches 0 from the right is 5/6, which, when rounded to the nearest hundredth, is approximately 0.83. This result tells us that as x gets infinitesimally close to 0 from the positive side, the value of the function f(x) gets increasingly close to 0.83. This understanding of the function's behavior near a specific point is a cornerstone of calculus and has far-reaching implications in various fields. In real-world applications, limits often help us analyze and predict the behavior of systems as certain parameters approach critical values. For example, in physics, limits are used to determine the instantaneous velocity or acceleration of an object, even when the time interval becomes infinitesimally small. In engineering, limits are crucial for analyzing the stability of structures or the performance of control systems. The concept of limits also extends beyond numerical values. We can talk about limits of sequences, limits of functions at infinity, and even limits in more abstract mathematical spaces. The versatility and power of limits make them an indispensable tool for mathematicians, scientists, and engineers alike. The ability to precisely describe and understand the behavior of functions as their inputs approach specific values is essential for solving a wide range of problems and gaining deeper insights into the world around us. In this particular example, the limit of 0.83 represents a specific point on the graph of the function f(x). It indicates the height of the function as x gets arbitrarily close to 0 from the right, providing a crucial piece of information about the function's behavior in that region.

H3: Final Answer

The limiting value as x approaches 0 from the right is approximately 0.83.