Finding The Limit Of (sqrt(x+h)-sqrt(x))/h As H Approaches 0

In calculus, the concept of limits is fundamental. Limits allow us to analyze the behavior of functions as their input approaches a particular value. One common application of limits is in the definition of the derivative, which measures the instantaneous rate of change of a function. In this article, we will explore how to find the limit of quotients as h approaches 0, focusing on the specific example: $\frac{\sqrt{x+h}-\sqrt{x}}{h}$.

This type of limit often appears in the context of finding derivatives using the limit definition. Understanding how to manipulate and evaluate these limits is crucial for mastering calculus. We will walk through the steps involved in finding this limit, highlighting the techniques and concepts used along the way. Specifically, we'll focus on the technique of rationalizing the numerator, a common strategy for handling limits involving square roots.

Before diving into the specifics of the quotient, let's briefly discuss the concept of a limit. A limit describes the value that a function approaches as its input gets closer and closer to a certain value. Mathematically, we write this as:

$$\lim_{x \to a} f(x) = L$$

This means that as x gets arbitrarily close to a, the function f(x) gets arbitrarily close to L. The limit does not necessarily have to be the value of the function at x = a; it's about the behavior of the function near a. Limits are the backbone of calculus, forming the basis for concepts like continuity, derivatives, and integrals.

When evaluating limits, we often encounter indeterminate forms, such as 0/0, which don't immediately tell us the value of the limit. In these cases, we need to use algebraic manipulation or other techniques to rewrite the expression in a form where the limit can be easily determined. One such technique is rationalizing the numerator or denominator, which is particularly useful when dealing with expressions involving square roots. In this article, our main focus is on mastering these techniques through a detailed example.

We aim to find the limit of the following quotient as h approaches 0:

$$\lim_{h \to 0} \frac{\sqrt{x+h}-\sqrt{x}}{h}$$

This expression represents the difference quotient, which is a key component in the definition of the derivative. The challenge here is that if we directly substitute h = 0, we get the indeterminate form 0/0. This means we need to manipulate the expression algebraically to eliminate this indeterminacy. The standard technique for dealing with such expressions involving square roots is to rationalize the numerator.

Rationalizing the numerator involves multiplying both the numerator and the denominator by the conjugate of the numerator. The conjugate of ${\sqrt{x+h}-\sqrt{x}}$ is ${\sqrt{x+h}+\sqrt{x}}$. This process eliminates the square roots in the numerator, making the expression easier to simplify and evaluate the limit. This technique is not just a trick; it's a systematic way to rewrite the expression to reveal its underlying behavior as h approaches 0. Let's proceed step-by-step to see how this works in practice.

To rationalize the numerator, we multiply both the numerator and the denominator by the conjugate of the numerator:

$$\lim_{h \to 0} \frac{\sqrt{x+h}-\sqrt{x}}{h} \cdot \frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}$$

Multiplying the numerators, we use the difference of squares formula, which states that (a - b)(a + b) = a² - b²:

$$\lim_{h \to 0} \frac{(\sqrt{x+h})^2 - (\sqrt{x})^2}{h(\sqrt{x+h}+\sqrt{x})}$$

This simplifies to:

$$\lim_{h \to 0} \frac{(x+h) - x}{h(\sqrt{x+h}+\sqrt{x})}$$

Now, we can simplify the numerator by canceling out the x terms:

$$\lim_{h \to 0} \frac{h}{h(\sqrt{x+h}+\sqrt{x})}$$

We can now cancel out the h in the numerator and denominator, provided that h is not equal to 0. Since we are taking the limit as h approaches 0, we are not concerned with the value of the expression at h = 0, but rather its behavior as h gets arbitrarily close to 0. Thus, we can safely cancel the h terms:

$$\lim_{h \to 0} \frac{1}{\sqrt{x+h}+\sqrt{x}}$$

Now that we have simplified the expression, we can directly substitute h = 0 to evaluate the limit:

$$\lim_{h \to 0} \frac{1}{\sqrt{x+h}+\sqrt{x}} = \frac{1}{\sqrt{x+0}+\sqrt{x}}$$

This simplifies to:

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

Thus, the limit of the quotient as h approaches 0 is:

$$\frac{1}{2\sqrt{x}}$$

We have successfully found the limit of the given quotient as h approaches 0. The key technique we used was rationalizing the numerator, which allowed us to eliminate the indeterminate form 0/0 and simplify the expression. This process involved multiplying the numerator and denominator by the conjugate of the numerator, applying the difference of squares formula, and then canceling out the common factor h. Understanding these steps is crucial for tackling similar limit problems in calculus. The final result, ${\frac{1}{2\sqrt{x}}}$, represents the derivative of the function ${\sqrt{x}}$ with respect to x, illustrating the connection between limits and derivatives.

This example highlights the importance of algebraic manipulation in evaluating limits. Direct substitution is often the first step, but when it leads to an indeterminate form, we need to employ techniques like rationalization to rewrite the expression. The ability to identify and apply appropriate algebraic techniques is a fundamental skill in calculus and is essential for solving a wide range of problems. As we've seen, mastering limits is not just about memorizing rules; it's about developing a deep understanding of how functions behave and how to manipulate expressions to reveal that behavior.

This particular limit is also significant because it demonstrates the derivative of the square root function. In calculus, the derivative represents the instantaneous rate of change of a function, and finding derivatives is a core concept. By understanding how to find limits like this, we build a strong foundation for understanding more advanced calculus topics.

Limits, Calculus, Rationalizing the Numerator, Indeterminate Form, Derivative, Difference Quotient, Conjugate, Algebraic Manipulation, Square Root, Function, Evaluate Limits, Quotients, h approaches 0, Instantaneous Rate of Change, Simplifying Expressions, Techniques for Limits, Mathematical Concepts, Understanding Limits, Solving Limit Problems