Evaluate The Following Limit:B. $\lim _{x \rightarrow \infty} \frac{6x^4 - 3x^3 + 2x^2 + 4}{6x^3 + 1}$

Introduction

Limits are a fundamental concept in calculus, and evaluating them is crucial in understanding various mathematical concepts. In this article, we will focus on evaluating the limit of a rational function, specifically the limit as x approaches infinity of the given function. We will break down the process into manageable steps, making it easier to understand and apply.

Understanding Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given function, we have a polynomial in the numerator and another polynomial in the denominator. Rational functions can be evaluated using various techniques, including factoring, canceling, and using limit properties.

The Given Function

The given function is:

$\lim _{x \rightarrow \infty} \frac{6x^4 - 3x^3 + 2x^2 + 4}{6x^3 + 1}$

Step 1: Divide Both the Numerator and Denominator by the Highest Power of x

To evaluate the limit, we need to divide both the numerator and denominator by the highest power of x, which is $x^3$. This will help us simplify the expression and make it easier to evaluate.

$\lim _{x \rightarrow \infty} \frac{\frac{6x^4}{x^3} - \frac{3x^3}{x^3} + \frac{2x^2}{x^3} + \frac{4}{x^3}}{\frac{6x^3}{x^3} + \frac{1}{x^3}}$

Step 2: Simplify the Expression

After dividing both the numerator and denominator by $x^3$, we can simplify the expression.

$\lim _{x \rightarrow \infty} \frac{6x - 3 + \frac{2}{x} + \frac{4}{x^3}}{6 + \frac{1}{x^3}}$

Step 3: Evaluate the Limit

Now that we have simplified the expression, we can evaluate the limit as x approaches infinity.

As x approaches infinity, the terms $\frac{2}{x}$ and $\frac{4}{x^3}$ approach 0, and the term $\frac{1}{x^3}$ also approaches 0.

$\lim _{x \rightarrow \infty} \frac{6x - 3 + 0 + 0}{6 + 0}$

Step 4: Simplify the Expression

After evaluating the limit, we can simplify the expression.

$\lim _{x \rightarrow \infty} \frac{6x - 3}{6}$

Step 5: Evaluate the Limit

Now that we have simplified the expression, we can evaluate the limit as x approaches infinity.

As x approaches infinity, the term $6x$ approaches infinity, and the term $-3$ approaches a constant value.

$\lim _{x \rightarrow \infty} \frac{\infty}{6}$

Conclusion

The limit of the given function as x approaches infinity is infinity.

Final Answer

The final answer is $\boxed{\infty}$.

Why is this Important?

Evaluating limits of rational functions is crucial in understanding various mathematical concepts, including calculus and algebra. It is also essential in solving problems in physics, engineering, and other fields. By following the steps outlined in this article, you can evaluate limits of rational functions with ease.

Common Mistakes to Avoid

When evaluating limits of rational functions, it is essential to avoid common mistakes, including:

• Not dividing both the numerator and denominator by the highest power of x
• Not simplifying the expression after dividing
• Not evaluating the limit as x approaches infinity

Tips and Tricks

When evaluating limits of rational functions, here are some tips and tricks to keep in mind:

• Always divide both the numerator and denominator by the highest power of x
• Simplify the expression after dividing
• Evaluate the limit as x approaches infinity
• Use limit properties, such as the sum and product rules, to simplify the expression

Real-World Applications

Evaluating limits of rational functions has numerous real-world applications, including:

• Physics: Evaluating limits of rational functions is essential in solving problems in physics, such as calculating the trajectory of a projectile or the motion of an object under the influence of gravity.
• Engineering: Evaluating limits of rational functions is crucial in designing and analyzing systems, such as electrical circuits or mechanical systems.
• Economics: Evaluating limits of rational functions is essential in understanding economic concepts, such as supply and demand or the behavior of markets.

Conclusion

Evaluating limits of rational functions is a crucial concept in mathematics, and it has numerous real-world applications. By following the steps outlined in this article, you can evaluate limits of rational functions with ease. Remember to avoid common mistakes and use limit properties to simplify the expression. With practice and patience, you can become proficient in evaluating limits of rational functions.

Introduction

Evaluating limits of rational functions is a crucial concept in mathematics, and it has numerous real-world applications. In our previous article, we provided a step-by-step guide on how to evaluate limits of rational functions. In this article, we will provide a Q&A guide to help you better understand the concept and address any questions or concerns you may have.

Q: What is a rational function?

A: A rational function is a function that can be expressed as the ratio of two polynomials. It is a function that has a numerator and a denominator, and the numerator and denominator are both polynomials.

Q: What is the highest power of x?

A: The highest power of x is the highest power of x that appears in the numerator or denominator of the rational function. For example, in the rational function $\frac{x^2 + 3x + 2}{x + 1}$, the highest power of x is $x^2$.

Q: Why do we need to divide both the numerator and denominator by the highest power of x?

A: We need to divide both the numerator and denominator by the highest power of x to simplify the expression and make it easier to evaluate the limit. By doing so, we can eliminate the highest power of x and make the expression more manageable.

Q: What happens to the terms $\frac{2}{x}$ and $\frac{4}{x^3}$ as x approaches infinity?

A: As x approaches infinity, the terms $\frac{2}{x}$ and $\frac{4}{x^3}$ approach 0. This is because as x gets larger and larger, the denominator gets larger and larger, and the fraction gets smaller and smaller.

Q: What is the final answer to the given function?

A: The final answer to the given function is $\boxed{\infty}$.

Q: Why is evaluating limits of rational functions important?

A: Evaluating limits of rational functions is important because it has numerous real-world applications, including physics, engineering, and economics. It is also essential in understanding various mathematical concepts, including calculus and algebra.

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:

• Not dividing both the numerator and denominator by the highest power of x
• Not simplifying the expression after dividing
• Not evaluating the limit as x approaches infinity

Q: What are some tips and tricks for evaluating limits of rational functions?

A: Some tips and tricks for evaluating limits of rational functions include:

• Always divide both the numerator and denominator by the highest power of x
• Simplify the expression after dividing
• Evaluate the limit as x approaches infinity
• Use limit properties, such as the sum and product rules, to simplify the expression

Q: Can you provide some examples of real-world applications of evaluating limits of rational functions?

A: Yes, here are some examples of real-world applications of evaluating limits of rational functions:

• Physics: Evaluating limits of rational functions is essential in solving problems in physics, such as calculating the trajectory of a projectile or the motion of an object under the influence of gravity.
• Engineering: Evaluating limits of rational functions is crucial in designing and analyzing systems, such as electrical circuits or mechanical systems.
• Economics: Evaluating limits of rational functions is essential in understanding economic concepts, such as supply and demand or the behavior of markets.

Conclusion

Evaluating limits of rational functions is a crucial concept in mathematics, and it has numerous real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in evaluating limits of rational functions. Remember to use limit properties and simplify the expression to make it easier to evaluate the limit. With practice and patience, you can become a master of evaluating limits of rational functions.