Evaluating Limits Using L'Hôpital's Rule Solve Lim X To 0 Sin(14x) Tan(10x)

Introduction

In the realm of calculus, evaluating limits is a fundamental skill. Limits help us understand the behavior of functions as they approach specific values, and they form the bedrock of concepts like continuity and derivatives. Among the various techniques for limit evaluation, L'Hôpital's Rule stands out as a powerful tool for handling indeterminate forms. This article delves into the application of L'Hôpital's Rule to solve the limit $\lim _{x \rightarrow 0} \frac{\sin (14 x)}{\tan (10 x)}$. We'll break down the process step by step, ensuring a clear understanding of the underlying principles and techniques. Mastering L'Hôpital's Rule not only enhances your problem-solving capabilities but also provides a deeper appreciation for the elegance and power of calculus. Understanding and applying L'Hôpital's Rule is crucial for tackling more complex calculus problems. The rule itself is relatively straightforward: if you encounter an indeterminate form like 0/0 or ∞/∞, you can differentiate the numerator and the denominator separately and then re-evaluate the limit. This process can be repeated as many times as necessary until the limit can be determined. However, it's essential to verify that the conditions for applying L'Hôpital's Rule are met before using it. This involves confirming that the limit results in an indeterminate form and that the derivatives of the numerator and denominator exist. In this article, we will thoroughly explore these conditions and demonstrate how to apply L'Hôpital's Rule effectively to find the exact value of the given limit. By the end of this guide, you will have a comprehensive understanding of how to approach similar limit problems and apply this powerful calculus tool with confidence.

Understanding the Problem: $\lim _{x \rightarrow 0} \frac{\sin (14 x)}{\tan (10 x)}$

Before diving into the solution, it's crucial to understand the problem we're tackling. The limit we're trying to evaluate is $\lim _{x \rightarrow 0} \frac{\sin (14 x)}{\tan (10 x)}$. This means we want to determine what value the function $\frac{\sin (14 x)}{\tan (10 x)}$ approaches as x gets arbitrarily close to 0. Directly substituting x = 0 into the function gives us $\frac{\sin (14 \cdot 0)}{\tan (10 \cdot 0)} = \frac{\sin (0)}{\tan (0)} = \frac{0}{0}$, which is an indeterminate form. Indeterminate forms are expressions in calculus that don't have a definitive value and require further analysis to resolve. The most common indeterminate forms include 0/0, ∞/∞, 0 \cdot ∞, ∞ − ∞, 1^∞, 0^0, and ∞^0. These forms arise because the limit of a quotient, product, difference, or power depends on the relative rates at which the functions involved approach their respective limits. In the case of 0/0, both the numerator and the denominator are approaching zero, so the overall limit is not immediately obvious. Similarly, for ∞/∞, both the numerator and denominator are growing without bound, making the limit ambiguous. The presence of an indeterminate form signals that we cannot simply plug in the limit value to find the answer. Instead, we need to employ techniques such as algebraic manipulation, trigonometric identities, or, in this case, L'Hôpital's Rule to simplify the expression and reveal the true limit. Recognizing the indeterminate form is the first critical step in solving limit problems, as it dictates the appropriate methods to use. This particular problem, with its 0/0 indeterminate form, is a classic candidate for L'Hôpital's Rule, which we will explore in detail in the following sections. By carefully applying this rule, we can transform the complex limit into a more manageable form, ultimately leading us to the exact value.

Verifying the Indeterminate Form and Conditions for L'Hôpital's Rule

As highlighted earlier, the first step in applying L'Hôpital's Rule is to verify that the limit results in an indeterminate form. In our case, as $x$ approaches 0, both $\sin(14x)$ and $\tan(10x)$ approach 0. This gives us the indeterminate form 0/0, which satisfies the primary condition for using L'Hôpital's Rule. However, there are additional conditions that must be met to ensure the correct application of the rule. L'Hôpital's Rule states that if the limit $\lim_{x \to c} \frac{f(x)}{g(x)}$ results in an indeterminate form (0/0 or ∞/∞), and if the following conditions are satisfied:

1. Both $f(x)$ and $g(x)$ are differentiable on an open interval containing c, except possibly at c itself.
2. $g'(x) \neq 0$ on this interval, except possibly at c.
3. The limit $\lim_{x \to c} \frac{f'(x)}{g'(x)}$ exists, then $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$.

Let's verify these conditions for our problem. We have $f(x) = \sin(14x)$ and $g(x) = \tan(10x)$.

1. Differentiability: Both $\sin(14x)$ and $\tan(10x)$ are differentiable functions for all x in an open interval around 0, except where $\tan(10x)$ is undefined (which occurs at multiples of $\frac{\pi}{20}$, but not at 0). So, this condition is satisfied.
2. Non-zero derivative: We need to check that $g'(x) \neq 0$ near 0. The derivative of $g(x) = \tan(10x)$ is $g'(x) = 10\sec^2(10x)$. As x approaches 0, $\sec^2(10x)$ approaches 1, so $g'(x)$ approaches 10, which is not zero. Thus, this condition is also satisfied.
3. Existence of the limit of derivatives: We will determine whether the limit of the derivatives exists after we differentiate the numerator and the denominator. This is the crux of applying L'Hôpital's Rule – finding the derivatives and re-evaluating the limit. By meticulously checking these conditions, we ensure that we are applying L'Hôpital's Rule correctly and that the result will be valid. Neglecting to verify these conditions can lead to incorrect conclusions, so it is an essential step in the process. Now that we have confirmed the applicability of L'Hôpital's Rule, we can proceed with differentiating the functions and evaluating the new limit.

Applying L'Hôpital's Rule: Differentiating and Re-evaluating

With the conditions for L'Hôpital's Rule verified, we can now proceed to apply the rule. This involves differentiating both the numerator and the denominator of our function, and then re-evaluating the limit. Our original function is $\frac{\sin(14x)}{\tan(10x)}$. Let's find the derivatives of the numerator and the denominator separately.

• The derivative of the numerator, $f(x) = \sin(14x)$, is $f'(x) = 14\cos(14x)$. This is a straightforward application of the chain rule in differentiation.
• The derivative of the denominator, $g(x) = \tan(10x)$, is $g'(x) = 10\sec^2(10x)$. Again, this uses the chain rule along with the derivative of the tangent function.

Now, according to L'Hôpital's Rule, we need to evaluate the limit of the ratio of these derivatives: $\lim_{x \rightarrow 0} \frac{f'(x)}{g'(x)} = \lim_{x \rightarrow 0} \frac{14\cos(14x)}{10\sec^2(10x)}$. At this point, we re-evaluate the limit by substituting x = 0 into the expression. This gives us:

$\frac{14\cos(14 \cdot 0)}{10\sec^2(10 \cdot 0)} = \frac{14\cos(0)}{10\sec^2(0)}$

Since $\cos(0) = 1$ and $\sec(0) = \frac{1}{\cos(0)} = 1$, we have:

$\frac{14 \cdot 1}{10 \cdot 1^2} = \frac{14}{10} = \frac{7}{5}$

Therefore, the limit of the ratio of the derivatives is $\frac{7}{5}$. This value represents the limit of our original function as x approaches 0, according to L'Hôpital's Rule. The process of differentiating and re-evaluating is a critical step in solving indeterminate forms. It transforms the original complex limit into a simpler one that can be directly evaluated. In this case, a single application of L'Hôpital's Rule was sufficient to resolve the indeterminate form and find the limit. However, in some cases, the limit of the derivatives may still be an indeterminate form, necessitating repeated applications of L'Hôpital's Rule until a determinate value is obtained. The key is to carefully differentiate and consistently re-evaluate until the limit can be found.

Final Result: $\lim _{x \rightarrow 0} \frac{\sin (14 x)}{\tan (10 x)} = \frac{7}{5}$

After meticulously applying L'Hôpital's Rule, we have arrived at the final result. We started with the limit $\lim _{x \rightarrow 0} \frac{\sin (14 x)}{\tan (10 x)}$, which presented an indeterminate form of 0/0. By differentiating both the numerator and the denominator and then re-evaluating the limit, we found that:

$\lim _{x \rightarrow 0} \frac{\sin (14 x)}{\tan (10 x)} = \lim _{x \rightarrow 0} \frac{14\cos(14x)}{10\sec^2(10x)} = \frac{7}{5}$

Therefore, the exact value of the limit is $\frac{7}{5}$. This result provides a precise understanding of the function's behavior as x approaches 0. The function $\frac{\sin (14 x)}{\tan (10 x)}$ gets closer and closer to $\frac{7}{5}$ as x gets arbitrarily close to 0. The journey to this solution has highlighted the power and utility of L'Hôpital's Rule in dealing with indeterminate forms. We've seen how differentiating the numerator and denominator can transform a complex limit into a simpler one that can be readily evaluated. This technique is invaluable in calculus and is used extensively in various fields of mathematics, science, and engineering. Understanding and applying L'Hôpital's Rule effectively requires careful attention to detail, including verifying the conditions for its application and correctly performing the differentiation. By mastering this rule, you enhance your ability to tackle a wide range of limit problems and deepen your understanding of calculus concepts. The result $\frac{7}{5}$ not only answers the specific problem but also serves as a testament to the elegance and precision of mathematical tools in uncovering the behavior of functions.

Conclusion

In conclusion, the evaluation of $\lim _{x \rightarrow 0} \frac{\sin (14 x)}{\tan (10 x)}$ demonstrates the effectiveness of L'Hôpital's Rule in handling indeterminate forms. We methodically verified the conditions for applying the rule, differentiated the numerator and denominator, and re-evaluated the limit to arrive at the exact value of $\frac{7}{5}$. This process underscores the importance of understanding the underlying principles of calculus and the proper application of its techniques. L'Hôpital's Rule is a powerful tool, but its correct usage depends on verifying the necessary conditions and performing accurate differentiation. This example serves as a valuable illustration of how to approach limit problems involving indeterminate forms, providing a clear and step-by-step methodology. By mastering this technique, students and practitioners can confidently tackle more complex calculus problems and gain a deeper appreciation for the beauty and precision of mathematical analysis. The ability to evaluate limits accurately is fundamental to many areas of mathematics and its applications, making L'Hôpital's Rule an indispensable tool in the calculus toolbox. Through careful application and a solid understanding of the concepts, we can unlock the behavior of functions and solve problems that initially appear intractable. The result $\frac{7}{5}$ is not just an answer; it is a demonstration of the power of calculus to reveal the hidden patterns and relationships within mathematical expressions.