Evaluate Limit As X Approaches Infinity Of (2x+1)/(x^2+x+1)

When evaluating limits at infinity of rational functions, understanding the behavior of polynomials as x grows without bound is crucial. A rational function is essentially a fraction where both the numerator and denominator are polynomials. Let's consider the function:

${ f(x) = \frac{2x + 1}{x^2 + x + 1} }$

Our goal is to find the limit of this function as x approaches infinity, which we write as:

${ \lim_{x \to \infty} \frac{2x + 1}{x^2 + x + 1} }$

Understanding the Behavior of Polynomials at Infinity

To effectively evaluate this limit, we must first understand how polynomials behave as x becomes extremely large. The term with the highest power in a polynomial dominates its behavior as x approaches infinity. For example, in the polynomial x² + x + 1, the x² term will significantly outweigh the x and the constant term as x gets larger and larger.

In the given function, the numerator has a highest power term of 2x, while the denominator has a highest power term of x². The degree of the numerator (1) is less than the degree of the denominator (2). This difference in degrees is key to determining the limit as x approaches infinity. When the degree of the denominator is greater than the degree of the numerator, the function will approach 0 as x goes to infinity.

Dividing by the Highest Power of x in the Denominator

A common technique to evaluate limits at infinity for rational functions involves dividing both the numerator and the denominator by the highest power of x that appears in the denominator. In our case, the highest power of x in the denominator is x². We divide both the numerator and the denominator by x²:

${ \lim_{x \to \infty} \frac{2x + 1}{x^2 + x + 1} = \lim_{x \to \infty} \frac{\frac{2x}{x^2} + \frac{1}{x^2}}{\frac{x^2}{x^2} + \frac{x}{x^2} + \frac{1}{x^2}} }$

Simplifying the fractions, we get:

${ \lim_{x \to \infty} \frac{\frac{2}{x} + \frac{1}{x^2}}{1 + \frac{1}{x} + \frac{1}{x^2}} }$

Now, we can analyze each term as x approaches infinity.

Evaluating the Limit Term by Term

As x approaches infinity, terms of the form 1/xⁿ, where n is a positive integer, approach 0. This is because dividing a constant by an increasingly large number results in a value that gets closer and closer to zero. Applying this to our simplified expression:

• ${\frac{2}{x}}$ approaches 0 as x approaches infinity.
• ${\frac{1}{x^2}}$ approaches 0 as x approaches infinity.
• ${\frac{1}{x}}$ approaches 0 as x approaches infinity.

Substituting these limits into our expression, we get:

${ \lim_{x \to \infty} \frac{\frac{2}{x} + \frac{1}{x^2}}{1 + \frac{1}{x} + \frac{1}{x^2}} = \frac{0 + 0}{1 + 0 + 0} = \frac{0}{1} = 0 }$

Thus, the limit of the function as x approaches infinity is 0.

Conclusion: Limit Evaluation

In conclusion, the limit of the function ${ \frac{2x + 1}{x^2 + x + 1} }$ as x approaches infinity is 0. This result stems from the fact that the degree of the denominator is greater than the degree of the numerator. By dividing both the numerator and the denominator by the highest power of x in the denominator and then evaluating the limit term by term, we were able to determine the function's behavior as x grows infinitely large. This method provides a robust approach for evaluating limits of rational functions at infinity, ensuring accurate results by focusing on the dominant terms and their influence on the function's overall behavior. Understanding these principles is essential for tackling more complex limit problems and for applications in calculus and mathematical analysis.

To effectively evaluate the limit of the function ${ f(x) = \frac{2x + 1}{x^2 + x + 1} }$ as x approaches infinity, a step-by-step approach is essential. This method ensures clarity and accuracy in understanding the behavior of the function as x grows without bound. We will systematically break down the process, highlighting key techniques and concepts along the way. First, we must recognize that we are dealing with a rational function, which is a ratio of two polynomials. The behavior of such functions as x approaches infinity is largely determined by the degrees of these polynomials.

Step 1: Identify the Highest Power of x in the Denominator

Identifying the highest power of x in the denominator is the first crucial step. This term dominates the behavior of the denominator as x becomes very large. In our function, ${ f(x) = \frac{2x + 1}{x^2 + x + 1} }$, the denominator is x² + x + 1. The term with the highest power is x², so we will use this to normalize the expression. This step sets the stage for simplifying the function in a way that allows us to easily evaluate its limit at infinity. Recognizing the dominant term is pivotal in handling rational functions at infinity, as it dictates how we manipulate the expression to reveal its limiting behavior. By focusing on the x² term, we can effectively reduce the complexity of the problem and pave the way for a straightforward evaluation.

Step 2: Divide Both Numerator and Denominator by the Highest Power of x

Next, we divide both the numerator and the denominator by x², which is the highest power of x in the denominator. This step is crucial because it allows us to redistribute the terms and simplify the expression, making it easier to evaluate the limit. Doing this, we get:

${ \lim_{x \to \infty} \frac{2x + 1}{x^2 + x + 1} = \lim_{x \to \infty} \frac{\frac{2x}{x^2} + \frac{1}{x^2}}{\frac{x^2}{x^2} + \frac{x}{x^2} + \frac{1}{x^2}} }$

This transformation is based on the fundamental principle that dividing both the numerator and denominator of a fraction by the same non-zero value does not change the fraction's overall value. It’s a clever algebraic manipulation that sets the stage for evaluating the limit. By normalizing the expression in this way, we make the individual terms more manageable and reveal the underlying structure that determines the limit at infinity. This step is pivotal for transforming a complex rational function into a form where its limiting behavior can be readily discerned.

Step 3: Simplify the Expression

After dividing by x², we simplify the expression by reducing the fractions. This involves canceling out common factors and rewriting the terms in their simplest form. The simplified expression becomes:

${ \lim_{x \to \infty} \frac{\frac{2}{x} + \frac{1}{x^2}}{1 + \frac{1}{x} + \frac{1}{x^2}} }$

Simplification is a critical step because it transforms the original complicated fraction into one where the behavior of each term as x approaches infinity is clearer. The fractions now have x in the denominator, which makes it evident that these terms will approach zero as x becomes very large. This process of simplification highlights the dominant terms and facilitates a more intuitive understanding of the limit. By reducing the expression to its simplest form, we pave the way for the final evaluation, making the limiting behavior of the function transparent.

Step 4: Evaluate the Limit as x Approaches Infinity

Now we evaluate the limit as x approaches infinity. As x becomes very large, the terms ${\frac{2}{x}}$, ${\frac{1}{x^2}}$, and ${\frac{1}{x}}$ all approach 0. This is because dividing a constant by an increasingly large number results in a value that gets closer and closer to zero. Substituting these values into our simplified expression, we get:

${ \lim_{x \to \infty} \frac{\frac{2}{x} + \frac{1}{x^2}}{1 + \frac{1}{x} + \frac{1}{x^2}} = \frac{0 + 0}{1 + 0 + 0} = \frac{0}{1} = 0 }$

Thus, the limit of the function ${ f(x) = \frac{2x + 1}{x^2 + x + 1} }$ as x approaches infinity is 0. This result demonstrates how the denominator, which has a higher degree, dominates the behavior of the function as x grows without bound. Evaluating the limit at this stage involves understanding the fundamental concept of how fractions behave as their denominators become infinitely large. By substituting the limiting values of individual terms, we can accurately determine the overall limit of the function.

Step 5: Conclusion of Limit Evaluation

The final conclusion is that the limit of the function ${ \frac{2x + 1}{x^2 + x + 1} }$ as x approaches infinity is 0. This conclusion is derived from the systematic evaluation of the function, which included identifying the highest power of x in the denominator, dividing both the numerator and denominator by this term, simplifying the expression, and then evaluating the limit. The critical insight here is that when the degree of the denominator is greater than the degree of the numerator, the function approaches 0 as x goes to infinity. This methodical approach not only provides the answer but also enhances understanding of how rational functions behave at infinity. This principle is widely applicable in calculus and analysis, and mastering it is essential for tackling more complex limit problems. Understanding these steps ensures a clear and accurate evaluation, reinforcing the foundation for more advanced mathematical concepts. This step-by-step process not only helps in solving the problem but also builds a strong understanding of the underlying mathematical principles.

While dividing by the highest power of x in the denominator is a standard technique, alternative methods can provide additional insights and approaches for evaluating limits at infinity, especially for rational functions. Exploring these alternatives can deepen our understanding and offer flexibility in problem-solving. Two notable alternatives are the intuitive comparison of polynomial degrees and the application of L'Hôpital's Rule.

Method 1: Comparing Polynomial Degrees

One of the most straightforward ways to determine the limit of a rational function as x approaches infinity is by comparing the degrees of the polynomials in the numerator and denominator. The degree of a polynomial is the highest power of x in the polynomial. For instance, in the function ${ f(x) = \frac{2x + 1}{x^2 + x + 1} }$, the degree of the numerator (2x + 1) is 1, and the degree of the denominator (x² + x + 1) is 2.

There are three cases to consider when comparing the degrees:

1. If the degree of the numerator is less than the degree of the denominator, the limit as x approaches infinity is 0. This is the case in our example, where the degree of the numerator (1) is less than the degree of the denominator (2). Therefore, ${ \lim_{x \to \infty} \frac{2x + 1}{x^2 + x + 1} = 0 }$.
2. If the degree of the numerator is equal to the degree of the denominator, the limit as x approaches infinity is the ratio of the leading coefficients. The leading coefficient is the coefficient of the highest power term. For example, if we had the function ${ g(x) = \frac{3x^2 + 2x + 1}{2x^2 + x + 3} }$, the degrees are equal (both 2), and the limit is ${ \frac{3}{2} }$.
3. If the degree of the numerator is greater than the degree of the denominator, the limit as x approaches infinity is either infinity (${ \infty }$) or negative infinity (${ -\infty }$), depending on the signs of the leading coefficients and the behavior of the function as x becomes very large. For instance, with ${ h(x) = \frac{x^3 + 1}{x^2 + 1} }$, the limit is infinity.

This method provides a quick and intuitive way to determine the limit by focusing on the dominant terms in the numerator and denominator. It leverages the understanding that the highest degree terms dictate the function's behavior as x approaches infinity. By comparing these degrees, we can often find the limit without complex algebraic manipulations. This approach is particularly useful for quickly assessing the overall trend of a rational function at infinity.

Method 2: L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool for evaluating limits of indeterminate forms, which include ${\frac{0}{0}}$ and ${\frac{\infty}{\infty}}$. This rule states that if ${ \lim_{x \to c} \frac{f(x)}{g(x)} }$ is of an indeterminate form, then:

${ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} }$

provided the limit on the right side exists. Here, ${ f'(x) }$ and ${ g'(x) }$ are the derivatives of f(x) and g(x), respectively.

To apply L'Hôpital's Rule to our function ${ f(x) = \frac{2x + 1}{x^2 + x + 1} }$ as x approaches infinity, we first verify that it is of the form ${\frac{\infty}{\infty}}$. As x approaches infinity, both the numerator (2x + 1) and the denominator (x² + x + 1) approach infinity, so we can apply the rule.

We compute the derivatives of the numerator and the denominator:

• The derivative of 2x + 1 is 2.
• The derivative of x² + x + 1 is 2x + 1.

Applying L'Hôpital's Rule once, we get:

${ \lim_{x \to \infty} \frac{2x + 1}{x^2 + x + 1} = \lim_{x \to \infty} \frac{2}{2x + 1} }$

As x approaches infinity, the denominator (2x + 1) approaches infinity, while the numerator remains constant (2). Therefore, the limit is 0.

L'Hôpital's Rule is especially useful when dealing with more complex rational functions or when the degree comparison method is not immediately clear. It transforms the original limit into a simpler form by differentiating the numerator and the denominator. In some cases, it may be necessary to apply L'Hôpital's Rule multiple times to obtain a determinate form. However, it's crucial to ensure that the limit is indeed of an indeterminate form before applying the rule. This method provides a structured approach to handle challenging limit problems, making it a valuable tool in calculus.

Conclusion

In summary, while dividing by the highest power of x is a reliable method, comparing polynomial degrees and applying L'Hôpital's Rule offer alternative strategies for evaluating limits at infinity. The degree comparison method provides a quick assessment based on the dominant terms, while L'Hôpital's Rule offers a systematic approach for indeterminate forms. By mastering these techniques, you can tackle a wide range of limit problems more efficiently and deepen your understanding of the behavior of functions at infinity. Each method has its strengths and applications, and the choice of method may depend on the specific function and the desired level of detail in the evaluation. Understanding these alternatives enhances problem-solving skills and provides a broader perspective on limit evaluation in calculus.