Evaluating The Limit Of [√(x+h) - √x] / H A Calculus Exploration

In the realm of calculus, limits serve as a foundational concept, providing the bedrock upon which derivatives and integrals are built. The limit, in essence, describes the value a function approaches as its input gets closer and closer to a specific value. Today, we delve into a particularly intriguing limit problem: lim (h->0) [√(x+h) - √x] / h. This limit, at first glance, might appear perplexing, but it holds a significant connection to the concept of derivatives, specifically the derivative of the square root function. This article aims to provide a comprehensive exploration of this limit, elucidating the steps involved in its evaluation and highlighting its relevance in the broader context of calculus. Our exploration will involve algebraic manipulation, a touch of clever trickery, and a deep dive into the underlying principles of limits. So, fasten your seatbelts as we embark on this mathematical journey, unraveling the intricacies of this limit and gaining a deeper appreciation for the power of calculus.

Before we tackle the specific limit at hand, it's crucial to establish a solid understanding of the limit definition itself. In simple terms, the limit of a function f(x) as x approaches a value c (denoted as lim (x->c) f(x) ) represents the value that f(x) gets arbitrarily close to as x gets arbitrarily close to c, without necessarily equaling c. This concept is central to calculus and forms the basis for understanding continuity, derivatives, and integrals. The formal definition of a limit, often referred to as the epsilon-delta definition, provides a rigorous framework for this concept. It states that for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε, where L is the limit. While we won't delve into the epsilon-delta definition in detail here, understanding its essence is crucial for appreciating the rigor behind limit calculations. In essence, the limit allows us to analyze the behavior of functions at points where direct substitution might lead to undefined results, such as the case we are about to explore. The limit we are examining, lim (h->0) [√(x+h) - √x] / h, presents such a scenario, as direct substitution of h = 0 leads to an indeterminate form (0/0). Thus, we must employ alternative techniques to evaluate this limit.

The limit we aim to solve is lim (h→0) [√(x+h) - √x] / h. The initial temptation might be to directly substitute h = 0 into the expression. However, this approach immediately leads to a problem. Substituting h = 0 results in [√(x+0) - √x] / 0, which simplifies to (√x - √x) / 0, and further to 0/0. This outcome, 0/0, is an indeterminate form. Indeterminate forms, such as 0/0, ∞/∞, 0 * ∞, and ∞ - ∞, signal that the limit cannot be evaluated through direct substitution alone. These forms indicate a need for further algebraic manipulation or alternative techniques to unveil the true value the function approaches. The reason 0/0 is indeterminate stems from the fact that it doesn't provide enough information to determine the limit. It could potentially approach any value, or even diverge. The numerator and denominator are both approaching zero, but the rate at which they do so dictates the actual limit. In our case, the presence of the square roots adds another layer of complexity. Therefore, we must employ a strategy that bypasses the indeterminate form and reveals the hidden behavior of the function as h approaches zero. The most common and effective strategy for this type of limit is to rationalize the numerator, which we will explore in the next section.

To overcome the indeterminate form and evaluate the limit lim (h→0) [√(x+h) - √x] / h, we employ a powerful algebraic technique known as rationalizing the numerator. This technique involves multiplying both the numerator and the denominator of the expression by the conjugate of the numerator. The conjugate of √(x+h) - √x is √(x+h) + √x. Multiplying by the conjugate leverages the difference of squares identity, (a - b)(a + b) = a² - b², to eliminate the square roots in the numerator. Let's see this in action:

[ (√(x+h) - √x) / h ] * [ (√(x+h) + √x) / (√(x+h) + √x) ]

When we multiply the numerators, we get:

(√(x+h) - √x) * (√(x+h) + √x) = (√(x+h))² - (√x)² = (x + h) - x = h

The denominator becomes:

h * (√(x+h) + √x)

Now, our expression looks like this:

h / [ h * (√(x+h) + √x) ]

The beauty of this step lies in the fact that we can now cancel out the h in the numerator and denominator, provided that h ≠ 0 (which is valid since we are taking the limit as h approaches 0, not when h equals 0). This cancellation is a crucial step, as it removes the source of the indeterminate form. The act of rationalizing the numerator has transformed the expression into a form that is much easier to handle and evaluate as h approaches zero.

Following the rationalization of the numerator in the limit lim (h→0) [√(x+h) - √x] / h, we arrived at the expression h / [ h * (√(x+h) + √x) ]. The crucial step now is to simplify this expression by canceling out the common factor of h in the numerator and the denominator. As we discussed earlier, this cancellation is valid because we are considering the limit as h approaches 0, which means h is never actually equal to 0. This subtle distinction is paramount in limit calculations. Cancelling the h terms, we get:

1 / (√(x+h) + √x)

This simplified expression is significantly easier to work with than the original. The indeterminate form has been eliminated, and we are now in a position to directly substitute h = 0 without encountering any issues. The act of simplification has essentially revealed the true behavior of the function as h approaches zero. This highlights the importance of algebraic manipulation in limit evaluation. Often, a seemingly complex limit can be tamed through clever algebraic techniques, transforming it into a form that is amenable to direct substitution. In our case, rationalizing the numerator and simplifying the resulting expression have paved the way for a straightforward evaluation of the limit.

After successfully rationalizing the numerator and simplifying the expression, we are now poised to evaluate the limit lim (h→0) [√(x+h) - √x] / h. Our simplified expression is 1 / (√(x+h) + √x). At this stage, we can directly substitute h = 0 into the expression. This substitution yields:

1 / (√(x+0) + √x) = 1 / (√x + √x) = 1 / (2√x)

Therefore, the limit is:

lim (h→0) [√(x+h) - √x] / h = 1 / (2√x)

This result is quite significant. It reveals that the limit of the given expression as h approaches 0 is 1 / (2√x). This limit, as we will discuss later, has a profound connection to the derivative of the square root function. The process of evaluating this limit underscores the power of algebraic manipulation and simplification in calculus. By strategically employing techniques like rationalizing the numerator, we can transform seemingly intractable limits into forms that can be readily evaluated. The final result, 1 / (2√x), not only provides the answer to our specific limit problem but also hints at a deeper relationship between limits and derivatives.

The limit we have just evaluated, lim (h→0) [√(x+h) - √x] / h = 1 / (2√x), is not just an isolated mathematical curiosity; it has a deep connection to the concept of derivatives in calculus. In fact, this limit is the very definition of the derivative of the function f(x) = √x. The derivative of a function at a point represents the instantaneous rate of change of the function at that point, or, geometrically, the slope of the tangent line to the function's graph at that point. The formal definition of the derivative, often denoted as f'(x), is given by:

f'(x) = lim (h→0) [f(x+h) - f(x)] / h

Comparing this definition with the limit we solved, we can see a direct correspondence. In our case, f(x) = √x, so f(x+h) = √(x+h). Substituting these into the definition of the derivative, we get:

f'(x) = lim (h→0) [√(x+h) - √x] / h

This is precisely the limit we evaluated, and we found it to be 1 / (2√x). Therefore, we can conclude that the derivative of f(x) = √x is f'(x) = 1 / (2√x). This connection highlights the fundamental role of limits in defining derivatives. Limits provide the rigorous foundation for understanding instantaneous rates of change, which are central to many applications of calculus in physics, engineering, economics, and other fields. Our exploration of this limit has not only provided a specific answer but has also illuminated a key link between limits and the broader landscape of calculus.

In conclusion, we have successfully navigated the intricacies of the limit lim (h→0) [√(x+h) - √x] / h. We began by recognizing the challenge posed by direct substitution, which led to the indeterminate form 0/0. To overcome this hurdle, we employed the technique of rationalizing the numerator, a powerful algebraic tool that allowed us to eliminate the square roots in the numerator and simplify the expression. After simplification, we were able to directly substitute h = 0 and obtain the result 1 / (2√x). This result, however, is more than just a numerical answer; it unveils a profound connection to the concept of derivatives. We discovered that the limit we evaluated is, in fact, the very definition of the derivative of the square root function, f(x) = √x. This connection underscores the fundamental role of limits in calculus, serving as the foundation upon which derivatives and other advanced concepts are built. Our journey through this limit problem has not only provided a specific solution but has also deepened our understanding of the core principles of calculus and the power of algebraic manipulation in problem-solving. The ability to evaluate limits like this is crucial for understanding the behavior of functions and their rates of change, which are essential in various scientific and engineering disciplines. By mastering these techniques, we unlock the potential to explore more complex mathematical concepts and apply them to real-world problems.