Evaluating The Limit Of (sqrt(x^2+7x))/(12-4x) As X Approaches Infinity

This article delves into the fascinating realm of limits, specifically focusing on the evaluation of the limit $\lim _{x \rightarrow \infty} \frac{\sqrt{x^2+7 x}}{12-4 x}$. We will embark on a step-by-step journey, dissecting the problem, employing relevant mathematical techniques, and ultimately arriving at the solution. Understanding limits is crucial in calculus and real analysis, as it forms the bedrock for concepts such as continuity, derivatives, and integrals. This exploration will not only provide the answer but also illuminate the underlying principles that govern limit calculations, especially those involving infinity. The limit presented here is a classic example that showcases how algebraic manipulation and a keen understanding of dominant terms can lead to a clear and concise solution. Let's begin our exploration and uncover the intricacies of this limit. We will start by examining the behavior of the function as x approaches infinity, which is the core of limit evaluation. This involves understanding how the terms within the function interact and which terms become dominant as x grows without bound. We will then proceed to manipulate the expression algebraically, aiming to simplify it into a form where the limit can be easily determined. This often involves techniques such as dividing by the highest power of x present in the expression, a common strategy when dealing with limits at infinity. Throughout this process, we will emphasize the importance of rigorous mathematical reasoning and the careful application of limit laws. The goal is not just to arrive at the correct answer, but also to develop a deeper understanding of the concepts involved and the techniques used. By the end of this article, you will not only be able to solve this particular limit problem, but also gain a valuable toolkit for tackling a wider range of limit problems, especially those encountered in calculus and related fields. Remember, practice is key to mastering limits, and this article serves as a stepping stone in that journey. So, let's dive in and unravel the mysteries of this limit together. This journey promises to be both insightful and rewarding, enhancing your understanding of calculus and mathematical analysis.

Initial Assessment and Strategy

Before diving into the calculations, let's analyze the given limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{x^2+7 x}}{12-4 x}$. As x approaches infinity, we observe that both the numerator and the denominator also tend towards infinity. This results in an indeterminate form of type $\frac{\infty}{\infty}$. Indeterminate forms require careful manipulation before the limit can be evaluated. A common strategy for dealing with such forms, particularly when dealing with polynomials and radicals, is to divide both the numerator and the denominator by the highest power of x present in the expression. This allows us to isolate the dominant terms and simplify the expression. In this case, the highest power of x under the square root is $x^2$, which means we should consider dividing by x outside the square root. However, we must be cautious about how we handle the square root. Specifically, when x is approaching positive infinity, $\sqrt{x^2}$ is equal to |x|, which is equal to x. This is a crucial point to remember, as it affects how we simplify the expression. The denominator has a term of x raised to the power of 1. Therefore, we will divide both the numerator and the denominator by x. This step is essential for simplifying the limit and revealing its true value. By dividing by x, we are essentially normalizing the expression and making the dominant terms more apparent. This allows us to see how the function behaves as x grows infinitely large. The strategy of dividing by the highest power of x is a powerful technique in limit evaluation, particularly when dealing with rational functions and expressions involving radicals. It allows us to transform the indeterminate form into a form where the limit can be easily calculated. This technique relies on the fact that as x approaches infinity, the terms with the highest powers dominate the behavior of the function. By dividing by the highest power of x, we are essentially focusing on these dominant terms and eliminating the less significant terms. This simplification process is crucial for accurately evaluating the limit. In the next section, we will apply this strategy and perform the algebraic manipulations necessary to simplify the expression and ultimately determine the limit. So, let's proceed with the calculations and see how this strategy unfolds.

Algebraic Manipulation: Dividing by x

To evaluate the limit $\lim _{x \rightarrow \infty} \frac{\sqrt{x^2+7 x}}{12-4 x}$, we will divide both the numerator and the denominator by x. This is a standard technique for handling limits at infinity, especially when dealing with rational functions or expressions involving radicals. First, let's rewrite the expression inside the limit by dividing both the numerator and the denominator by x:

$\lim _{x \rightarrow \infty} \frac{\frac{\sqrt{x^2+7 x}}{x}}{\frac{12-4 x}{x}}$

Now, we need to carefully handle the x in the denominator of the numerator. Since x is approaching positive infinity, we can write x as $\sqrt{x^2}$. This allows us to bring the x inside the square root:

$\lim _{x \rightarrow \infty} \frac{\sqrt{\frac{x^2+7 x}{x^2}}}{\frac{12}{x}-4}$

Simplifying the expression inside the square root gives us:

$\lim _{x \rightarrow \infty} \frac{\sqrt{1+\frac{7}{x}}}{\frac{12}{x}-4}$

This step is crucial because it allows us to separate the terms that approach zero as x approaches infinity. The term $\frac{7}{x}$ in the numerator and the term $\frac{12}{x}$ in the denominator will both tend to zero as x becomes infinitely large. This simplification process is a key element in evaluating limits at infinity. By dividing by x and simplifying the expression, we have transformed the original limit into a form that is much easier to analyze. The next step is to apply the limit laws and directly evaluate the limit as x approaches infinity. This involves substituting the limit value into the simplified expression and determining the final result. The algebraic manipulation we have performed has set the stage for a straightforward evaluation of the limit. So, let's proceed to the final step and determine the value of the limit.

Evaluating the Limit

Now that we have simplified the expression, we can evaluate the limit $\lim _{x \rightarrow \infty} \frac{\sqrt{1+\frac{7}{x}}}{\frac{12}{x}-4}$. As x approaches infinity, the terms $\frac{7}{x}$ and $\frac{12}{x}$ approach 0. This is a fundamental concept in limit evaluation: any constant divided by a quantity that approaches infinity will approach zero. Substituting these limits into the expression, we get:

$\frac{\sqrt{1+0}}{0-4} = \frac{\sqrt{1}}{-4} = \frac{1}{-4}$

Therefore, the limit is:

$\lim _{x \rightarrow \infty} \frac{\sqrt{x^2+7 x}}{12-4 x} = -\frac{1}{4}$

This result indicates that as x becomes infinitely large, the function $\frac{\sqrt{x^2+7 x}}{12-4 x}$ approaches the value -1/4. This is a negative value, which is consistent with the observation that the denominator (12 - 4x) becomes increasingly negative as x approaches infinity, while the numerator remains positive. The process of evaluating this limit demonstrates the power of algebraic manipulation and the application of limit laws. By dividing by the highest power of x and simplifying the expression, we were able to transform the indeterminate form into a form where the limit could be easily determined. This technique is widely used in calculus and real analysis for evaluating limits at infinity. The final result, -1/4, is a precise value that represents the asymptotic behavior of the function as x grows without bound. This understanding of limits is crucial for various applications, including analyzing the behavior of functions, determining the convergence of sequences and series, and understanding the concepts of continuity and derivatives. In conclusion, the evaluation of this limit showcases the elegance and rigor of mathematical analysis, providing a valuable insight into the behavior of functions at infinity.

Conclusion

In this article, we successfully evaluated the limit $\lim _{x \rightarrow \infty} \frac{\sqrt{x^2+7 x}}{12-4 x}$. We began by recognizing the indeterminate form $\frac{\infty}{\infty}$ and then employed the strategy of dividing both the numerator and the denominator by x. This crucial step allowed us to simplify the expression and reveal the underlying behavior of the function as x approaches infinity. We carefully handled the square root, recognizing that $\sqrt{x^2}$ is equal to |x|, which is equal to x when x is approaching positive infinity. This is a subtle but important point that is essential for accurate limit evaluation. The algebraic manipulation led us to the simplified expression $\frac{\sqrt{1+\frac{7}{x}}}{\frac{12}{x}-4}$. From this form, we could easily see that the terms $\frac{7}{x}$ and $\frac{12}{x}$ approach 0 as x approaches infinity. Applying the limit laws, we arrived at the final answer of -1/4. This result signifies that the function $\frac{\sqrt{x^2+7 x}}{12-4 x}$ approaches -1/4 as x grows infinitely large. The process of evaluating this limit highlights the importance of understanding indeterminate forms and employing appropriate algebraic techniques to simplify expressions. The strategy of dividing by the highest power of x is a powerful tool in limit evaluation, particularly when dealing with rational functions and expressions involving radicals. Furthermore, this exercise reinforces the fundamental concepts of limits and their applications in calculus and real analysis. Limits are the foundation upon which many other concepts are built, including continuity, derivatives, and integrals. A solid understanding of limits is therefore essential for anyone pursuing studies in mathematics or related fields. This article serves as a valuable example of how to approach limit problems systematically and rigorously, providing a clear and concise solution along with a detailed explanation of the underlying principles. By mastering these techniques, you will be well-equipped to tackle a wide range of limit problems and further your understanding of calculus and mathematical analysis. Remember, practice is key, so continue to explore and solve different types of limit problems to solidify your skills.