Evaluating Limits With Radicals Solving Lim (x->a) (√(x+a) - √(3x-a))/(x-a)

Introduction

Hey guys! Today, we're diving into a fascinating problem from the realm of calculus: evaluating the limit ${\lim _{x \rightarrow a} \frac{\sqrt{x+a}-\sqrt{3 x-a}}{x-a}}$. This isn't your run-of-the-mill limit; it requires a bit of algebraic finesse and a solid understanding of limit properties. We'll break down the problem step-by-step, making sure everyone, from calculus newbies to seasoned math enthusiasts, can follow along. So, grab your thinking caps, and let's get started!

The limit problem presented here is a classic example of an indeterminate form. When we directly substitute x = a into the expression, we get a 0/0 situation, which doesn't tell us much. This is where the fun begins! We need to manipulate the expression to reveal the true behavior of the function as x approaches a. The key technique we'll employ is rationalization. Rationalization involves getting rid of square roots in either the numerator or the denominator by multiplying by a clever form of 1. In this case, we'll rationalize the numerator. This involves multiplying both the numerator and the denominator by the conjugate of the numerator, which is ${\sqrt{x+a} + \sqrt{3x-a}}$. This might seem like a magic trick, but it's a standard algebraic technique that allows us to simplify the expression and eliminate the indeterminate form. Understanding this method is crucial for tackling many limit problems involving radicals. So, let’s dive into the solution step-by-step and unravel this mathematical puzzle.

Step-by-Step Solution

1. The Indeterminate Form

First, let's confirm that we indeed have an indeterminate form. Substituting x = a into the expression, we get:

${ \frac{\sqrt{a+a} - \sqrt{3a-a}}{a-a} = \frac{\sqrt{2a} - \sqrt{2a}}{0} = \frac{0}{0} }$

Yep, it's the classic 0/0, signaling that we need to do some algebraic manipulation. This initial assessment is critical. Recognizing the indeterminate form is the first step in choosing the correct strategy. Many limit problems will present this challenge, and identifying it early on prevents you from going down unproductive paths. So, always start by plugging in the value to see what you get. If it’s anything other than a determinate form (like a number or infinity), further manipulation is required. In this scenario, the 0/0 form clearly indicates that we need to employ a technique such as rationalization or L'Hôpital's Rule. For this particular problem, rationalization is the more straightforward and elegant approach, which we will proceed with in the next step.

2. Rationalizing the Numerator

To rationalize the numerator, we multiply both the numerator and denominator by its conjugate, which is ${\sqrt{x+a} + \sqrt{3x-a}}$:

${ \lim _{x \rightarrow a} \frac{\sqrt{x+a}-\sqrt{3 x-a}}{x-a} \cdot \frac{\sqrt{x+a} + \sqrt{3x-a}}{\sqrt{x+a} + \sqrt{3x-a}} }$

This rationalization process is the heart of solving this limit. By multiplying by the conjugate, we’re essentially using the difference of squares identity in reverse. This allows us to eliminate the square roots in the numerator, which is the key to simplifying the expression. The conjugate is formed by simply changing the sign between the two terms in the numerator. Multiplying both the numerator and the denominator by the same expression ensures that we're not changing the value of the original expression, only its form. This is a crucial point to remember: we’re performing an algebraic manipulation, not altering the underlying mathematical relationship. This technique is not only applicable to limits but also to other areas of mathematics where simplifying expressions involving radicals is necessary. Mastering this skill will significantly enhance your problem-solving abilities.

3. Simplifying the Expression

Multiplying out the numerator, we get:

${ \lim _{x \rightarrow a} \frac{(x+a) - (3x-a)}{(x-a)(\sqrt{x+a} + \sqrt{3x-a})} }$

Simplifying the numerator further:

${ \lim _{x \rightarrow a} \frac{x + a - 3x + a}{(x-a)(\sqrt{x+a} + \sqrt{3x-a})} = \lim _{x \rightarrow a} \frac{-2x + 2a}{(x-a)(\sqrt{x+a} + \sqrt{3x-a})} }$

Now, we can factor out a -2 from the numerator:

${ \lim _{x \rightarrow a} \frac{-2(x-a)}{(x-a)(\sqrt{x+a} + \sqrt{3x-a})} }$

The simplification step is where the magic truly happens. The numerator, after expansion and simplification, reveals a factor of (x - a), which is the same factor that's causing the indeterminate form in the denominator. This is a common pattern in limit problems: the manipulation often uncovers a common factor that can be canceled out. Factoring out the -2 from the numerator is a crucial step in making this cancellation obvious. Always look for opportunities to factor expressions, as this can lead to significant simplifications. The goal here is to eliminate the term that's causing the 0/0 form, and by factoring and canceling, we're doing just that. This simplification process transforms the original complex expression into a much more manageable form, paving the way for direct substitution in the next step.

4. Canceling the Common Factor

We can now cancel the (x-a) term from the numerator and denominator:

${ \lim _{x \rightarrow a} \frac{-2}{\sqrt{x+a} + \sqrt{3x-a}} }$

Canceling the common factor (x - a) is a pivotal moment in solving the limit. This step effectively removes the source of the indeterminate form. By dividing both the numerator and denominator by (x - a), we are essentially “filling the hole” in the function at x = a. The resulting expression is now continuous at x = a, which means we can directly substitute x = a without encountering any issues. This cancellation highlights the importance of algebraic manipulation in limit evaluation. It transforms the problem from one where direct substitution leads to an undefined result to one where the limit can be easily found. The ability to identify and cancel common factors is a fundamental skill in calculus and will serve you well in a variety of mathematical contexts.

5. Evaluating the Limit

Finally, we substitute x = a into the simplified expression:

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

Assuming a > 0, we have our answer!

This final evaluation is the culmination of all our hard work. After simplifying the expression, we can now directly substitute x = a and obtain a meaningful result. This is the payoff for the algebraic manipulations we performed earlier. The assumption that a > 0 is important because it ensures that the square root is defined in the real number system. The result, -1/${\sqrt{2a}}$, represents the value that the function approaches as x gets arbitrarily close to a. This value provides valuable information about the behavior of the function near the point x = a. Direct substitution, whenever possible, is the most straightforward way to evaluate limits. This final step underscores the power of algebraic manipulation in transforming complex expressions into simpler forms that can be easily evaluated.

Conclusion

So, there you have it! We've successfully evaluated the limit ${\lim _{x \rightarrow a} \frac{\sqrt{x+a}-\sqrt{3 x-a}}{x-a}}$ by recognizing the indeterminate form, rationalizing the numerator, simplifying the expression, canceling common factors, and finally, substituting the value. This problem showcases the importance of algebraic techniques in calculus and how a systematic approach can help us solve seemingly complex problems. Keep practicing, and you'll become a limit-solving pro in no time!

This conclusion summarizes the entire process we’ve undertaken to solve the limit problem. We’ve walked through each step, highlighting the key techniques and concepts involved. This serves as a reminder of the overall strategy and the individual skills that are essential for tackling similar problems. Recognizing the indeterminate form, employing rationalization, simplifying expressions, and canceling common factors are all crucial tools in the calculus toolkit. The final substitution provides the answer, but the journey to get there is just as important. Each step builds upon the previous one, demonstrating the interconnectedness of mathematical concepts. By understanding this systematic approach, you'll be better equipped to handle a wide range of limit problems and other challenges in calculus and beyond. Practice is indeed the key to mastering these skills, so keep exploring and applying these techniques to different scenarios.