For The Function F ( X F(x F ( X ] Given Below, Evaluate Lim ⁡ X → ∞ F ( X \lim _{x \rightarrow \infty} F(x Lim X → ∞ ​ F ( X ] And Lim ⁡ X → − ∞ F ( X \lim _{x \rightarrow -\infty} F(x Lim X → − ∞ ​ F ( X ]. F ( X ) = 2 E X + 3 4 E X + 2 F(x)=\frac{2 E^x+3}{4 E^x+2} F ( X ) = 4 E X + 2 2 E X + 3 ​

Introduction

When dealing with limits of rational functions, we often encounter functions that involve exponential terms. In this article, we will explore the evaluation of limits of a rational function that contains an exponential term. Specifically, we will examine the function $f(x)=\frac{2 e^x+3}{4 e^x+2}$ and determine the values of $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x)$.

Understanding the Function

The given function $f(x)=\frac{2 e^x+3}{4 e^x+2}$ is a rational function that involves an exponential term $e^x$. The function has two terms in the numerator and two terms in the denominator. To evaluate the limits, we need to understand the behavior of the function as $x$ approaches infinity and negative infinity.

Evaluating the Limit as $x$ Approaches Infinity

To evaluate the limit as $x$ approaches infinity, we can use the following approach:

$\lim _{x \rightarrow \infty} f(x) = \lim _{x \rightarrow \infty} \frac{2 e^x+3}{4 e^x+2}$

As $x$ approaches infinity, the term $e^x$ dominates the numerator and denominator. Therefore, we can rewrite the function as:

$\lim _{x \rightarrow \infty} f(x) = \lim _{x \rightarrow \infty} \frac{2 e^x}{4 e^x} + \frac{3}{4 e^x}$

Simplifying the expression, we get:

$\lim _{x \rightarrow \infty} f(x) = \lim _{x \rightarrow \infty} \frac{1}{2} + \frac{3}{4 e^x}$

As $x$ approaches infinity, the term $\frac{3}{4 e^x}$ approaches zero. Therefore, the limit is:

$\lim _{x \rightarrow \infty} f(x) = \frac{1}{2}$

Evaluating the Limit as $x$ Approaches Negative Infinity

To evaluate the limit as $x$ approaches negative infinity, we can use a similar approach:

$\lim _{x \rightarrow -\infty} f(x) = \lim _{x \rightarrow -\infty} \frac{2 e^x+3}{4 e^x+2}$

As $x$ approaches negative infinity, the term $e^x$ approaches zero. Therefore, we can rewrite the function as:

$\lim _{x \rightarrow -\infty} f(x) = \lim _{x \rightarrow -\infty} \frac{2}{4} + \frac{3}{4 e^x}$

Simplifying the expression, we get:

$\lim _{x \rightarrow -\infty} f(x) = \lim _{x \rightarrow -\infty} \frac{1}{2} + \frac{3}{4 e^x}$

As $x$ approaches negative infinity, the term $\frac{3}{4 e^x}$ approaches zero. Therefore, the limit is:

$\lim _{x \rightarrow -\infty} f(x) = \frac{1}{2}$

Conclusion

In this article, we evaluated the limits of the rational function $f(x)=\frac{2 e^x+3}{4 e^x+2}$ as $x$ approaches infinity and negative infinity. We found that both limits approach the value $\frac{1}{2}$. This result demonstrates the importance of understanding the behavior of rational functions with exponential terms as $x$ approaches infinity and negative infinity.

Future Directions

This article provides a foundation for understanding the limits of rational functions with exponential terms. Future research could explore the evaluation of limits of more complex rational functions, including those with multiple exponential terms or rational terms. Additionally, the application of limit evaluation techniques to real-world problems, such as modeling population growth or chemical reactions, could provide valuable insights into the behavior of complex systems.

References

• [1] Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
• [2] Anton, H. (2017). Calculus: Early Transcendentals. John Wiley & Sons.
• [3] Edwards, C. H. (2019). Calculus: Early Transcendentals. Pearson Education.

Glossary

• Rational function: A function that can be expressed as the ratio of two polynomials.
• Exponential term: A term that involves an exponential function, such as $e^x$.
• Limit: A value that a function approaches as the input value approaches a certain point.
• Infinity: A mathematical concept that represents an unbounded or infinite value.
• Negative infinity: A mathematical concept that represents a value that approaches negative infinity.
  

Introduction

In our previous article, we explored the evaluation of limits of a rational function that contains an exponential term. In this article, we will address some of the most frequently asked questions related to evaluating limits of rational functions.

Q: What is the difference between a rational function and an exponential function?

A: A rational function is a function that can be expressed as the ratio of two polynomials, while an exponential function is a function that involves an exponential term, such as $e^x$.

Q: How do I evaluate the limit of a rational function as $x$ approaches infinity?

A: To evaluate the limit of a rational function as $x$ approaches infinity, you can use the following approach:

1. Divide both the numerator and denominator by the highest power of $x$ in the denominator.
2. Simplify the expression and cancel out any common factors.
3. Evaluate the limit as $x$ approaches infinity.

Q: What if the rational function has multiple exponential terms? How do I evaluate the limit?

A: If the rational function has multiple exponential terms, you can use the following approach:

1. Identify the dominant exponential term, which is the term with the highest power of $x$.
2. Divide both the numerator and denominator by the dominant exponential term.
3. Simplify the expression and cancel out any common factors.
4. Evaluate the limit as $x$ approaches infinity.

Q: How do I evaluate the limit of a rational function as $x$ approaches negative infinity?

A: To evaluate the limit of a rational function as $x$ approaches negative infinity, you can use a similar approach to evaluating the limit as $x$ approaches infinity. However, you will need to consider the behavior of the function as $x$ approaches negative infinity.

Q: What if the rational function has a rational term in the denominator? How do I evaluate the limit?

A: If the rational function has a rational term in the denominator, you can use the following approach:

1. Identify the rational term in the denominator.
2. Divide both the numerator and denominator by the rational term.
3. Simplify the expression and cancel out any common factors.
4. Evaluate the limit as $x$ approaches infinity or negative infinity.

Q: Can I use L'Hopital's Rule to evaluate the limit of a rational function?

A: Yes, you can use L'Hopital's Rule to evaluate the limit of a rational function. However, you will need to ensure that the function meets the conditions for L'Hopital's Rule, which are:

1. The function is in the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
2. The function is differentiable at the point of interest.
3. The derivative of the function is not equal to zero at the point of interest.

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

A: Some common mistakes to avoid when evaluating limits of rational functions include:

1. Not canceling out common factors in the numerator and denominator.
2. Not considering the behavior of the function as $x$ approaches infinity or negative infinity.
3. Not using L'Hopital's Rule when necessary.
4. Not simplifying the expression before evaluating the limit.

Conclusion

In this article, we addressed some of the most frequently asked questions related to evaluating limits of rational functions. We hope that this article has provided you with a better understanding of how to evaluate limits of rational functions and has helped you to avoid common mistakes.