Evaluate The Limit:${ \lim _{x \rightarrow 0} \frac{x \cdot 3 X}{3 X-1} }$

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. In this article, we will focus on evaluating the limit of a complex function, specifically the limit of the expression $\frac{x \cdot 3^x}{3^x-1}$ as $x$ approaches 0. This type of problem requires a deep understanding of mathematical concepts, including limits, exponents, and algebraic manipulation.

Understanding the Problem

The given expression is a rational function, which means it is the ratio of two functions. In this case, the numerator is $x \cdot 3^x$, and the denominator is $3^x-1$. As $x$ approaches 0, we need to determine the value of this expression. At first glance, this problem may seem daunting, but with a step-by-step approach, we can break it down into manageable parts.

Step 1: Simplifying the Expression

To evaluate the limit, we need to simplify the expression as much as possible. One way to do this is to factor out the common term $3^x$ from the numerator and denominator. This gives us:

$$\frac{x \cdot 3^x}{3^x-1} = \frac{3^x \cdot x}{3^x-1}$$

Step 2: Using L'Hopital's Rule

As $x$ approaches 0, the expression becomes indeterminate, meaning that it is neither a finite number nor infinity. In this case, we can use L'Hopital's rule, which states that if a limit is indeterminate, we can take the derivative of the numerator and denominator separately and then take the limit again.

Step 3: Differentiating the Numerator and Denominator

To apply L'Hopital's rule, we need to differentiate the numerator and denominator separately. The derivative of the numerator is:

$$\frac{d}{dx} (3^x \cdot x) = 3^x + x \cdot 3^x \cdot \ln(3)$$

The derivative of the denominator is:

$$\frac{d}{dx} (3^x-1) = 3^x \cdot \ln(3)$$

Step 4: Evaluating the Limit

Now that we have the derivatives of the numerator and denominator, we can evaluate the limit again. This gives us:

$$\lim_{x \to 0} \frac{3^x + x \cdot 3^x \cdot \ln(3)}{3^x \cdot \ln(3)}$$

Step 5: Simplifying the Expression

We can simplify the expression further by canceling out the common term $3^x$ in the numerator and denominator. This gives us:

$$\lim_{x \to 0} \frac{1 + x \cdot \ln(3)}{\ln(3)}$$

Step 6: Evaluating the Limit

Finally, we can evaluate the limit by substituting $x=0$ into the expression. This gives us:

$$\frac{1 + 0 \cdot \ln(3)}{\ln(3)} = \frac{1}{\ln(3)}$$

Conclusion

In this article, we evaluated the limit of the expression $\frac{x \cdot 3^x}{3^x-1}$ as $x$ approaches 0. We used a step-by-step approach, including simplifying the expression, using L'Hopital's rule, and evaluating the limit. The final answer is $\frac{1}{\ln(3)}$, which is a fundamental concept in mathematics.

Applications of the Limit

The limit we evaluated in this article has several applications in mathematics and other fields. For example, it can be used to model population growth, chemical reactions, and electrical circuits. Additionally, it can be used to derive important mathematical formulas, such as the formula for the derivative of the natural logarithm function.

Future Research Directions

There are several future research directions that can be explored based on the limit we evaluated in this article. For example, we can investigate the behavior of the limit as $x$ approaches other values, such as infinity or negative infinity. We can also explore the applications of the limit in other fields, such as physics, engineering, and economics.

Conclusion

In conclusion, evaluating the limit of a complex function requires a deep understanding of mathematical concepts, including limits, exponents, and algebraic manipulation. By breaking down the problem into manageable parts and using techniques such as L'Hopital's rule, we can evaluate the limit and derive important mathematical formulas. The limit we evaluated in this article has several applications in mathematics and other fields, and there are several future research directions that can be explored based on this concept.

Introduction

In our previous article, we evaluated the limit of the expression $\frac{x \cdot 3^x}{3^x-1}$ as $x$ approaches 0. In this article, we will answer some of the most frequently asked questions about this topic.

Q: What is the limit of the expression $\frac{x \cdot 3^x}{3^x-1}$ as $x$ approaches 0?

A: The limit of the expression $\frac{x \cdot 3^x}{3^x-1}$ as $x$ approaches 0 is $\frac{1}{\ln(3)}$.

Q: Why do we need to use L'Hopital's rule to evaluate the limit?

A: We need to use L'Hopital's rule because the expression $\frac{x \cdot 3^x}{3^x-1}$ becomes indeterminate as $x$ approaches 0. L'Hopital's rule allows us to take the derivative of the numerator and denominator separately and then take the limit again.

Q: What is L'Hopital's rule?

A: L'Hopital's rule is a mathematical technique used to evaluate limits of indeterminate forms. It states that if a limit is indeterminate, we can take the derivative of the numerator and denominator separately and then take the limit again.

Q: How do we apply L'Hopital's rule to the expression $\frac{x \cdot 3^x}{3^x-1}$?

A: To apply L'Hopital's rule, we need to differentiate the numerator and denominator separately. The derivative of the numerator is $3^x + x \cdot 3^x \cdot \ln(3)$, and the derivative of the denominator is $3^x \cdot \ln(3)$.

Q: What is the derivative of the numerator and denominator?

A: The derivative of the numerator is $3^x + x \cdot 3^x \cdot \ln(3)$, and the derivative of the denominator is $3^x \cdot \ln(3)$.

Q: How do we simplify the expression after applying L'Hopital's rule?

A: After applying L'Hopital's rule, we can simplify the expression by canceling out the common term $3^x$ in the numerator and denominator. This gives us $\frac{1 + x \cdot \ln(3)}{\ln(3)}$.

Q: What is the final answer to the limit?

A: The final answer to the limit is $\frac{1}{\ln(3)}$.

Q: What are some of the applications of the limit?

A: The limit we evaluated in this article has several applications in mathematics and other fields. For example, it can be used to model population growth, chemical reactions, and electrical circuits. Additionally, it can be used to derive important mathematical formulas, such as the formula for the derivative of the natural logarithm function.

Q: What are some of the future research directions in this area?

A: There are several future research directions that can be explored based on the limit we evaluated in this article. For example, we can investigate the behavior of the limit as $x$ approaches other values, such as infinity or negative infinity. We can also explore the applications of the limit in other fields, such as physics, engineering, and economics.

Conclusion

In this article, we answered some of the most frequently asked questions about the limit of the expression $\frac{x \cdot 3^x}{3^x-1}$ as $x$ approaches 0. We hope that this article has provided a clear understanding of the concept and its applications. If you have any further questions, please don't hesitate to ask.