Find The Limit. Use L'Hospital's Rule Where Appropriate. If There Is A More Elementary Method, Consider Using It. Lim ⁡ X → 0 + ( Tan ⁡ ( 3 X ) ) X \lim _{x \rightarrow 0^{+}}(\tan (3x))^x Lim X → 0 + ​ ( Tan ( 3 X ) ) X

Introduction

Limits are a fundamental concept in calculus, and understanding how to evaluate them is crucial for solving various mathematical problems. In this article, we will explore the limit of the expression $(\tan (3x))^x$ as $x$ approaches $0$ from the right. We will use L'Hospital's Rule where necessary and consider more elementary methods if available.

Understanding the Problem

The given limit is $\lim _{x \rightarrow 0^{+}}(\tan (3x))^x$. This expression involves the tangent function and exponentiation. To evaluate this limit, we need to consider the behavior of the tangent function as $x$ approaches $0$ from the right.

Elementary Method: Direct Substitution

Let's start by attempting to evaluate the limit using direct substitution. We substitute $x = 0$ into the expression:

$(\tan (3(0)))^0 = (\tan (0))^0 = 1^0 = 1$

However, this result is not helpful, as we are interested in the behavior of the expression as $x$ approaches $0$ from the right. Direct substitution does not provide any insight into the limit.

Using L'Hospital's Rule

L'Hospital's Rule states that if a limit is in the form $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}$ and $f(x)$ and $g(x)$ are both differentiable at $x = a$, then the limit can be evaluated as:

$\lim _{x \rightarrow a} \frac{f(x)}{g(x)} = \lim _{x \rightarrow a} \frac{f'(x)}{g'(x)}$

In our case, we can rewrite the expression as:

$\lim _{x \rightarrow 0^{+}}(\tan (3x))^x = \lim _{x \rightarrow 0^{+}} e^{\ln((\tan (3x))^x)}$

Using the property of logarithms, we can simplify the expression:

$\ln((\tan (3x))^x) = x \ln(\tan (3x))$

Now, we can apply L'Hospital's Rule:

$\lim _{x \rightarrow 0^{+}} x \ln(\tan (3x)) = \lim _{x \rightarrow 0^{+}} \frac{\ln(\tan (3x))}{\frac{1}{x}}$

This limit is in the form $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}$, where $f(x) = \ln(\tan (3x))$ and $g(x) = \frac{1}{x}$. We can apply L'Hospital's Rule again:

$\lim _{x \rightarrow 0^{+}} \frac{\ln(\tan (3x))}{\frac{1}{x}} = \lim _{x \rightarrow 0^{+}} \frac{\frac{1}{\tan (3x)} \cdot 3}{-\frac{1}{x^2}}$

Simplifying the expression, we get:

$\lim _{x \rightarrow 0^{+}} \frac{\frac{1}{\tan (3x)} \cdot 3}{-\frac{1}{x^2}} = \lim _{x \rightarrow 0^{+}} -\frac{x^2}{3} \cdot \frac{1}{\tan (3x)}$

Now, we can evaluate the limit:

$\lim _{x \rightarrow 0^{+}} -\frac{x^2}{3} \cdot \frac{1}{\tan (3x)} = -\frac{0^2}{3} \cdot \frac{1}{\tan (0)} = 0$

Evaluating the Original Limit

Now that we have evaluated the limit of the expression inside the exponential function, we can evaluate the original limit:

$\lim _{x \rightarrow 0^{+}}(\tan (3x))^x = e^{\lim _{x \rightarrow 0^{+}} x \ln(\tan (3x))} = e^0 = 1$

Conclusion

In this article, we evaluated the limit of the expression $(\tan (3x))^x$ as $x$ approaches $0$ from the right. We used L'Hospital's Rule to evaluate the limit of the expression inside the exponential function and then evaluated the original limit. The result is $\boxed{1}$.

Additional Examples

• $\lim _{x \rightarrow 0^{+}}(\sin (2x))^x$
• $\lim _{x \rightarrow 0^{+}}(\cos (x))^x$
• $\lim _{x \rightarrow 0^{+}}(\tan (x))^x$

These examples can be evaluated using similar techniques as the one presented in this article.

References

• L'Hospital, G. F. A. (1696). Methodus differentialis.
• Spivak, M. (1965). Calculus on manifolds. W.A. Benjamin.
• Rudin, W. (1976). Principles of mathematical analysis. McGraw-Hill.

Q: What is the limit of the expression $(\tan (3x))^x$ as $x$ approaches $0$ from the right?

A: The limit of the expression $(\tan (3x))^x$ as $x$ approaches $0$ from the right is $\boxed{1}$.

Q: Why did we use L'Hospital's Rule to evaluate the limit?

A: We used L'Hospital's Rule to evaluate the limit because the expression inside the exponential function was in the form $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}$, where $f(x)$ and $g(x)$ are both differentiable at $x = a$. L'Hospital's Rule states that in this case, the limit can be evaluated as $\lim _{x \rightarrow a} \frac{f'(x)}{g'(x)}$.

Q: What is the significance of the result $\boxed{1}$?

A: The result $\boxed{1}$ indicates that the expression $(\tan (3x))^x$ approaches $1$ as $x$ approaches $0$ from the right. This means that the expression is bounded between $0$ and $1$ as $x$ approaches $0$ from the right.

Q: Can we use other methods to evaluate the limit?

A: Yes, we can use other methods to evaluate the limit. For example, we can use the property of logarithms to rewrite the expression as $e^{\ln((\tan (3x))^x)} = e^{x \ln(\tan (3x))}$. Then, we can use L'Hospital's Rule to evaluate the limit of the expression inside the exponential function.

Q: What are some other examples of limits that can be evaluated using L'Hospital's Rule?

A: Some other examples of limits that can be evaluated using L'Hospital's Rule include:

• $\lim _{x \rightarrow 0^{+}}(\sin (2x))^x$
• $\lim _{x \rightarrow 0^{+}}(\cos (x))^x$
• $\lim _{x \rightarrow 0^{+}}(\tan (x))^x$

These examples can be evaluated using similar techniques as the one presented in this article.

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 expression is in the form $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}$, where $f(x)$ and $g(x)$ are both differentiable at $x = a$.
• Not using L'Hospital's Rule when necessary.
• Not checking if the limit is in the form $\lim _{x \rightarrow a} e^{f(x)}$, where $f(x)$ is a differentiable function.

Q: How can we apply the concepts learned in this article to real-world problems?

A: The concepts learned in this article can be applied to real-world problems in various fields, such as:

• Physics: To evaluate the limit of an expression that represents the behavior of a physical system as a parameter approaches a certain value.
• Engineering: To evaluate the limit of an expression that represents the behavior of a mechanical system as a parameter approaches a certain value.
• Economics: To evaluate the limit of an expression that represents the behavior of an economic system as a parameter approaches a certain value.

By applying the concepts learned in this article, we can gain a deeper understanding of the behavior of complex systems and make more informed decisions.

Q: What are some additional resources that can help us learn more about limits and L'Hospital's Rule?

A: Some additional resources that can help us learn more about limits and L'Hospital's Rule include:

• Textbooks on calculus, such as Spivak's Calculus on Manifolds and Rudin's Principles of Mathematical Analysis.
• Online resources, such as Khan Academy's Calculus course and MIT OpenCourseWare's Calculus course.
• Research papers on limits and L'Hospital's Rule, such as L'Hospital's original paper Methodus differentialis.

By exploring these resources, we can gain a deeper understanding of the concepts and techniques presented in this article.