Evaluate The Limit:$\[ \lim _{x \rightarrow \theta} \frac{x \tan \theta-\theta \tan X}{x-\theta} \\]

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Introduction

In mathematics, evaluating limits is a crucial concept that helps us understand the behavior of functions as the input values approach a specific point. The given limit, $\lim _{x \rightarrow \theta} \frac{x \tan \theta-\theta \tan x}{x-\theta}$, involves trigonometric functions and requires careful analysis to determine its value. In this article, we will delve into the world of limits and explore the steps to evaluate this particular limit.

Understanding the Limit

Before we dive into the evaluation process, let's break down the given limit and understand its components. The limit is of the form $\lim _{x \rightarrow \theta} f(x)$, where $f(x) = \frac{x \tan \theta-\theta \tan x}{x-\theta}$. Here, $\theta$ is a fixed value, and $x$ is the variable that approaches $\theta$ as the limit is evaluated.

Applying L'Hopital's Rule

One of the most powerful tools for evaluating limits is L'Hopital's Rule, which states that if a limit is of the form $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}$ and both $f(x)$ and $g(x)$ approach $0$ or $\infty$ as $x$ approaches $a$, then the limit can be evaluated by taking the derivatives of $f(x)$ and $g(x)$ and finding the limit of the ratio of the derivatives.

In our case, the limit is of the form $\lim _{x \rightarrow \theta} \frac{x \tan \theta-\theta \tan x}{x-\theta}$, which can be rewritten as $\lim _{x \rightarrow \theta} \frac{\tan \theta - \frac{\theta}{x} \tan x}{1 - \frac{\theta}{x}}$. As $x$ approaches $\theta$, both the numerator and denominator approach $0$, so we can apply L'Hopital's Rule.

Evaluating the Derivatives

To apply L'Hopital's Rule, we need to find the derivatives of the numerator and denominator. Let's start with the numerator:

$$\frac{d}{dx} (x \tan \theta - \theta \tan x) = \tan \theta - \theta \sec^2 x \tan x$$

And the derivative of the denominator is:

$$\frac{d}{dx} (x - \theta) = 1$$

Applying L'Hopital's Rule Again

Now that we have the derivatives, we can apply L'Hopital's Rule again to evaluate the limit. We have:

$$\lim _{x \rightarrow \theta} \frac{\tan \theta - \theta \sec^2 x \tan x}{1} = \tan \theta - \theta \sec^2 \theta \tan \theta$$

Simplifying the Expression

The expression we obtained is still not in its simplest form. We can simplify it by using the trigonometric identity $\sec^2 x = 1 + \tan^2 x$. Substituting this into the expression, we get:

$$\tan \theta - \theta \sec^2 \theta \tan \theta = \tan \theta - \theta (1 + \tan^2 \theta) \tan \theta$$

Final Simplification

Now, let's simplify the expression further by distributing the $\tan \theta$ term:

$$\tan \theta - \theta (1 + \tan^2 \theta) \tan \theta = \tan \theta - \theta \tan \theta - \theta \tan^3 \theta$$

Combining Like Terms

We can combine the like terms to get:

$$\tan \theta - \theta \tan \theta - \theta \tan^3 \theta = \tan \theta (1 - \theta - \theta \tan^2 \theta)$$

Final Answer

After simplifying the expression, we can see that the limit evaluates to:

$$\lim _{x \rightarrow \theta} \frac{x \tan \theta-\theta \tan x}{x-\theta} = \tan \theta (1 - \theta - \theta \tan^2 \theta)$$

Conclusion

In this article, we evaluated the limit $\lim _{x \rightarrow \theta} \frac{x \tan \theta-\theta \tan x}{x-\theta}$ using L'Hopital's Rule and trigonometric identities. We broke down the limit into smaller components, applied L'Hopital's Rule, and simplified the expression to obtain the final answer. This process demonstrates the importance of careful analysis and simplification in evaluating limits.

Additional Resources

For more information on limits and L'Hopital's Rule, we recommend the following resources:

• L'Hopital's Rule on Wikipedia
• Limits on Khan Academy
• Trigonometric Identities on Mathway

Final Thoughts

Evaluating limits is a crucial concept in mathematics, and L'Hopital's Rule is a powerful tool for solving these types of problems. By breaking down the limit into smaller components and applying trigonometric identities, we can simplify the expression and obtain the final answer. We hope this article has provided a clear and concise explanation of how to evaluate the limit $\lim _{x \rightarrow \theta} \frac{x \tan \theta-\theta \tan x}{x-\theta}$.

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Introduction

In our previous article, we evaluated the limit $\lim _{x \rightarrow \theta} \frac{x \tan \theta-\theta \tan x}{x-\theta}$ using L'Hopital's Rule and trigonometric identities. In this article, we will answer some frequently asked questions about the limit and provide additional insights into the evaluation process.

Q: What is L'Hopital's Rule?

A: L'Hopital's Rule is a mathematical technique used to evaluate limits of the form $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}$ where both $f(x)$ and $g(x)$ approach $0$ or $\infty$ as $x$ approaches $a$. The rule states that the limit can be evaluated by taking the derivatives of $f(x)$ and $g(x)$ and finding the limit of the ratio of the derivatives.

Q: Why do we need to apply L'Hopital's Rule in this case?

A: We need to apply L'Hopital's Rule because the limit $\lim _{x \rightarrow \theta} \frac{x \tan \theta-\theta \tan x}{x-\theta}$ is of the form $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}$ where both $f(x)$ and $g(x)$ approach $0$ as $x$ approaches $\theta$. By applying L'Hopital's Rule, we can simplify the expression and evaluate the limit.

Q: What is the significance of the trigonometric identity $\sec^2 x = 1 + \tan^2 x$?

A: The trigonometric identity $\sec^2 x = 1 + \tan^2 x$ is used to simplify the expression $\tan \theta - \theta \sec^2 \theta \tan \theta$. By substituting this identity into the expression, we can simplify it further and obtain the final answer.

Q: Can we evaluate the limit without using L'Hopital's Rule?

A: While it is possible to evaluate the limit without using L'Hopital's Rule, it would be much more complicated and would likely involve more advanced trigonometric identities. L'Hopital's Rule provides a powerful tool for evaluating limits of this form, and it is often the most efficient way to obtain the final answer.

Q: What are some common mistakes to avoid when evaluating limits?

A: Some common mistakes to avoid when evaluating limits include:

• Not checking if the limit is of the form $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}$ where both $f(x)$ and $g(x)$ approach $0$ or $\infty$ as $x$ approaches $a$.
• Not applying L'Hopital's Rule when necessary.
• Not simplifying the expression using trigonometric identities.
• Not checking if the final answer is valid.

Q: What are some real-world applications of limits?

A: Limits have many real-world applications, including:

• Physics: Limits are used to describe the behavior of physical systems as certain parameters approach specific values.
• Engineering: Limits are used to design and optimize systems, such as electronic circuits and mechanical systems.
• Economics: Limits are used to model economic systems and make predictions about future trends.
• Computer Science: Limits are used to analyze the performance of algorithms and data structures.

Conclusion

In this article, we answered some frequently asked questions about the limit $\lim _{x \rightarrow \theta} \frac{x \tan \theta-\theta \tan x}{x-\theta}$ and provided additional insights into the evaluation process. We hope this article has been helpful in understanding the concept of limits and how to evaluate them using L'Hopital's Rule and trigonometric identities.

Additional Resources

For more information on limits and L'Hopital's Rule, we recommend the following resources:

• L'Hopital's Rule on Wikipedia
• Limits on Khan Academy
• Trigonometric Identities on Mathway

Final Thoughts

Evaluating limits is a crucial concept in mathematics, and L'Hopital's Rule is a powerful tool for solving these types of problems. By breaking down the limit into smaller components and applying trigonometric identities, we can simplify the expression and obtain the final answer. We hope this article has provided a clear and concise explanation of how to evaluate the limit $\lim _{x \rightarrow \theta} \frac{x \tan \theta-\theta \tan x}{x-\theta}$.