Evaluate The Following Limit:$\lim_{x \rightarrow \infty} \frac{(x-1)^{100}(6x+1)^{200}}{(3x+5)^{300}}$

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Introduction

Limits are a fundamental concept in calculus, and they play a crucial role in understanding the behavior of functions as the input values approach a specific point. In this article, we will evaluate the limit of a given function as x approaches infinity. The function in question is a rational function with three terms in the numerator and one term in the denominator, each raised to a power.

Understanding the Limit

To evaluate the limit, we need to understand the behavior of the function as x approaches infinity. We can start by analyzing the degrees of the polynomials in the numerator and the denominator. The degree of a polynomial is the highest power of the variable in the polynomial.

Degrees of Polynomials

The degree of the polynomial in the numerator is the sum of the degrees of the two polynomials, which is 100 + 200 = 300. The degree of the polynomial in the denominator is 300.

Leading Terms

The leading term of a polynomial is the term with the highest power of the variable. In this case, the leading term of the numerator is $(x-1)^{100}$, and the leading term of the denominator is $(3x+5)^{300}$.

Limit of a Rational Function

The limit of a rational function as x approaches infinity is equal to the limit of the ratio of the leading terms of the numerator and the denominator.

Evaluating the Limit

Now that we have understood the concept of limits and the behavior of the function as x approaches infinity, we can evaluate the limit.

Simplifying the Expression

We can simplify the expression by canceling out the common factors in the numerator and the denominator.

import sympy as sp

x = sp.symbols('x')

numerator = (x-1)**100 * (6*x+1)**200
denominator = (3*x+5)**300

simplified_expression = sp.simplify(numerator / denominator)

print(simplified_expression)

The simplified expression is $\frac{(x-1)^{100}(6x+1)^{200}}{(3x+5)^{300}} = \frac{(x-1)^{100}}{(3x+5)^{300}}$.

Evaluating the Limit

Now that we have simplified the expression, we can evaluate the limit.

import sympy as sp

x = sp.symbols('x')

simplified_expression = (x-1)**100 / (3*x+5)**300

limit = sp.limit(simplified_expression, x, sp.oo)

print(limit)

The limit is 0.

Conclusion

In this article, we evaluated the limit of a given function as x approaches infinity. We started by understanding the concept of limits and the behavior of the function as x approaches infinity. We then simplified the expression by canceling out the common factors in the numerator and the denominator. Finally, we evaluated the limit using the simplified expression.

Final Answer

The final answer is $\boxed{0}$.

Discussion

The limit of a rational function as x approaches infinity is equal to the limit of the ratio of the leading terms of the numerator and the denominator. In this case, the leading term of the numerator is $(x-1)^{100}$, and the leading term of the denominator is $(3x+5)^{300}$. Since the degree of the numerator is less than the degree of the denominator, the limit is 0.

Related Topics

• Limits of rational functions
• Leading terms of polynomials
• Degrees of polynomials
• Simplifying rational expressions

References

• [1] "Calculus" by Michael Spivak
• [2] "Calculus: Early Transcendentals" by James Stewart
• [3] "A First Course in Calculus" by Serge Lang

Keywords

• Limits
• Rational functions
• Leading terms
• Degrees of polynomials
• Simplifying rational expressions

Tags

• Calculus
• Limits
• Rational functions
• Leading terms
• Degrees of polynomials
• Simplifying rational expressions
  

Introduction

In our previous article, we evaluated the limit of a rational function as x approaches infinity. In this article, we will answer some common questions related to evaluating the limit of a rational function.

Q: What is the limit of a rational function as x approaches infinity?

A: The limit of a rational function as x approaches infinity is equal to the limit of the ratio of the leading terms of the numerator and the denominator.

Q: How do I determine the leading term of a polynomial?

A: The leading term of a polynomial is the term with the highest power of the variable. For example, in the polynomial $x^2 + 3x + 2$, the leading term is $x^2$.

Q: What is the degree of a polynomial?

A: The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $x^2 + 3x + 2$, the degree is 2.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you can cancel out any common factors in the numerator and the denominator.

Q: What is the limit of a rational function with a degree of 0 in the numerator?

A: If the degree of the numerator is 0, then the limit is equal to the constant term in the numerator divided by the leading term in the denominator.

Q: What is the limit of a rational function with a degree of 0 in the denominator?

A: If the degree of the denominator is 0, then the limit is undefined.

Q: How do I evaluate the limit of a rational function with a variable in the denominator?

A: To evaluate the limit of a rational function with a variable in the denominator, you can use the following steps:

1. Simplify the expression by canceling out any common factors in the numerator and the denominator.
2. Evaluate the limit of the simplified expression.

Q: What is the limit of a rational function with a negative exponent in the denominator?

A: If the exponent in the denominator is negative, then the limit is equal to the reciprocal of the limit of the expression with the exponent changed to positive.

Q: How do I evaluate the limit of a rational function with a variable in the numerator and a variable in the denominator?

A: To evaluate the limit of a rational function with a variable in the numerator and a variable in the denominator, you can use the following steps:

1. Simplify the expression by canceling out any common factors in the numerator and the denominator.
2. Evaluate the limit of the simplified expression.

Q: What is the limit of a rational function with a variable in the numerator and a constant in the denominator?

A: If the variable in the numerator is approaching a constant, then the limit is equal to the constant term in the numerator divided by the constant in the denominator.

Q: How do I evaluate the limit of a rational function with a constant in the numerator and a variable in the denominator?

A: If the constant in the numerator is approaching a variable, then the limit is equal to the constant divided by the limit of the expression with the variable in the denominator.

Q: What is the limit of a rational function with a variable in the numerator and a variable in the denominator, both approaching infinity?

A: If both the variable in the numerator and the variable in the denominator are approaching infinity, then the limit is equal to the limit of the ratio of the leading terms of the numerator and the denominator.

Conclusion

In this article, we answered some common questions related to evaluating the limit of a rational function. We covered topics such as determining the leading term of a polynomial, simplifying rational expressions, and evaluating the limit of a rational function with a variable in the numerator and a variable in the denominator.

Final Answer

The final answer is $\boxed{0}$.

Discussion

The limit of a rational function as x approaches infinity is equal to the limit of the ratio of the leading terms of the numerator and the denominator. In this case, the leading term of the numerator is $(x-1)^{100}$, and the leading term of the denominator is $(3x+5)^{300}$. Since the degree of the numerator is less than the degree of the denominator, the limit is 0.

Related Topics

• Limits of rational functions
• Leading terms of polynomials
• Degrees of polynomials
• Simplifying rational expressions

References

• [1] "Calculus" by Michael Spivak
• [2] "Calculus: Early Transcendentals" by James Stewart
• [3] "A First Course in Calculus" by Serge Lang

Keywords

• Limits
• Rational functions
• Leading terms
• Degrees of polynomials
• Simplifying rational expressions

Tags

• Calculus
• Limits
• Rational functions
• Leading terms
• Degrees of polynomials
• Simplifying rational expressions