Evaluate The Limit Of (√(x+3x^2))/(2x-1) As X Approaches Infinity

In the realm of calculus, evaluating limits is a fundamental skill. Limits allow us to understand the behavior of functions as their input approaches a specific value, including infinity. This article delves into the process of evaluating a particular limit problem, providing a step-by-step explanation and insights into the underlying mathematical principles. Specifically, we will explore how to find the limit of the function $\frac{\sqrt{x+3x^2}}{2x-1}$ as $x$ approaches infinity. This type of problem often appears in introductory calculus courses and serves as a good example of how to manipulate and simplify expressions involving radicals and polynomials to determine their asymptotic behavior.

Understanding Limits at Infinity

Before diving into the specific problem, it's crucial to grasp the concept of limits at infinity. When we say $x$ approaches infinity (denoted as $x \rightarrow \infty$), we are considering the function's behavior as $x$ becomes arbitrarily large. In the context of rational functions (functions that are ratios of polynomials) and functions involving radicals, this often involves identifying the dominant terms – the terms that grow the fastest as $x$ increases. These dominant terms dictate the function's overall behavior as $x$ tends towards infinity.

In our case, we have a function that combines a square root and a rational expression. The presence of the square root adds a layer of complexity, as we need to consider how the square root function affects the growth rate. The general strategy for evaluating limits of this type involves algebraic manipulation to isolate the dominant terms and simplify the expression. This might involve dividing both the numerator and denominator by a suitable power of $x$, a technique we will apply in the subsequent steps. Furthermore, understanding the properties of limits is essential. For example, the limit of a quotient is the quotient of the limits (provided the limit of the denominator is not zero), and the limit of a constant times a function is the constant times the limit of the function. These properties allow us to break down complex limits into simpler components.

Problem Statement: $\lim _{x \rightarrow \infty} \frac{\sqrt{x+3 x^2}}{2 x-1}$

Our goal is to evaluate the limit:

$$\lim _{x \rightarrow \infty} \frac{\sqrt{x+3 x^2}}{2 x-1}$$

This limit involves a rational function where the numerator contains a square root. As $x$ approaches infinity, both the numerator and denominator grow without bound. To determine the limit, we need to analyze the relative growth rates of the numerator and denominator. The key strategy here is to divide both the numerator and the denominator by a suitable power of $x$. We need to choose the power of $x$ that will simplify the expression and allow us to clearly see the dominant terms. The highest power of $x$ inside the square root is $x^2$, so the square root will behave like $x$ as $x$ approaches infinity. Therefore, we will divide both the numerator and the denominator by $x$. This will allow us to compare the coefficients of the leading terms and determine the limit. The choice of dividing by $x$ is crucial because it effectively normalizes the growth rates, allowing us to compare the leading coefficients and determine the limit. Understanding why we choose $x$ instead of $x^2$ or a constant is vital for mastering these types of limit problems.

Step-by-Step Solution

1. Divide the numerator and the denominator by $x$:

To divide the numerator by $x$, we need to bring $x$ inside the square root. Since $x = \sqrt{x^2}$ for $x > 0$, we have:

$$\frac{\sqrt{x+3 x^2}}{x} = \frac{\sqrt{x+3 x^2}}{\sqrt{x^2}} = \sqrt{\frac{x+3 x^2}{x^2}} = \sqrt{\frac{x}{x^2} + \frac{3 x^2}{x^2}} = \sqrt{\frac{1}{x} + 3}$$

Dividing the denominator by $x$ is straightforward:

$$\frac{2x - 1}{x} = 2 - \frac{1}{x}$$

Therefore, our limit becomes:

$$\lim _{x \rightarrow \infty} \frac{\sqrt{x+3 x^2}}{2 x-1} = \lim _{x \rightarrow \infty} \frac{\sqrt{\frac{1}{x} + 3}}{2 - \frac{1}{x}}$$

This step is pivotal as it transforms the original expression into a form where the behavior as x approaches infinity is more apparent. By dividing by $x$, we have effectively normalized the terms, making it easier to identify the dominant components.

2. Evaluate the limit as $x$ approaches infinity:

Now, we can evaluate the limit of the simplified expression. As $x$ approaches infinity, $\frac{1}{x}$ approaches 0. Thus, we have:

$$\lim _{x \rightarrow \infty} \frac{1}{x} = 0$$

Substituting this into our expression:

$$\lim _{x \rightarrow \infty} \frac{\sqrt{\frac{1}{x} + 3}}{2 - \frac{1}{x}} = \frac{\sqrt{0 + 3}}{2 - 0} = \frac{\sqrt{3}}{2}$$

This step highlights the importance of understanding the behavior of basic functions as their input approaches infinity. The fact that $\frac{1}{x}$ approaches 0 as $x$ approaches infinity is a fundamental concept in calculus and is crucial for evaluating limits of this type.

Final Result

Therefore, the limit is:

$$\lim _{x \rightarrow \infty} \frac{\sqrt{x+3 x^2}}{2 x-1} = \frac{\sqrt{3}}{2}$$

Conclusion

We have successfully evaluated the limit of the given function as $x$ approaches infinity. The key to solving this problem was to divide both the numerator and denominator by $x$, which allowed us to simplify the expression and identify the dominant terms. This technique is a standard approach for evaluating limits of rational functions and functions involving radicals at infinity. Understanding the concept of dominant terms and the behavior of basic functions like $\frac{1}{x}$ as $x$ approaches infinity is crucial for mastering limit problems. This example demonstrates the power of algebraic manipulation and the importance of understanding the underlying mathematical principles in calculus.

The process we followed illustrates a common strategy for dealing with limits at infinity. By identifying the highest power of $x$ and dividing by that power, we can effectively compare the growth rates of the different terms in the expression. This approach is applicable to a wide range of limit problems and is a fundamental tool in calculus. Remember, practice is key to mastering these techniques. Working through various examples will solidify your understanding and build your problem-solving skills.

Further Exploration

To further enhance your understanding of limits at infinity, consider exploring the following:

• More complex functions: Try evaluating limits of functions involving trigonometric functions, exponential functions, or logarithmic functions.
• L'Hôpital's Rule: Learn about L'Hôpital's Rule, which provides a powerful technique for evaluating limits of indeterminate forms.
• Applications of Limits: Investigate how limits are used in other areas of calculus, such as finding derivatives and integrals.

By continuing to explore these concepts, you will develop a deeper appreciation for the power and beauty of calculus. Remember that calculus is a building block for many areas of science and engineering, so mastering these fundamental concepts is essential for future success.