Question

Find the limits. $$ \lim _{n \rightarrow \infty} \frac{n}{2 n+1} $$

Short answer

The limit is \( \frac{1}{2} \).

Step-by-step solution

Identify the Type of Limit

We are given the limit \( \lim _{n \rightarrow \infty} \frac{n}{2n+1} \). This is a limit of a rational function as \( n \) approaches infinity. The degrees of the numerator and the denominator are both 1, which suggests that the limit can be determined by considering the leading coefficients.

Simplify the Expression

To find the limit, divide every term in the fraction by \( n \), the highest power of \( n \) present in the denominator. The expression becomes: \( \frac{n/n}{(2n)/n + 1/n} = \frac{1}{2 + \frac{1}{n}} \).

Evaluate the Simplified Expression as \( n \rightarrow \infty \)

As \( n \) approaches infinity, \( \frac{1}{n} \) approaches 0. Substituting this into the simplified expression we get: \( \frac{1}{2 + 0} = \frac{1}{2} \).

Confirm the Result

Re-examine each step to ensure accuracy. The terms simplify correctly, and the approach of \( n \rightarrow \infty \) leading \( \frac{1}{n} \) to 0 correctly gives the final fraction as \( \frac{1}{2} \). Therefore, the limit is correctly calculated as \( \frac{1}{2} \).

Key concepts

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. In simpler terms, you can think of them as fractions where both the numerator and the denominator are polynomials. For instance, in the problem at hand, the expression \( \frac{n}{2n+1} \) represents a rational function because both 'n' and '2n + 1' are polynomials. Rational functions are common in calculus, and they can appear in a variety of situations, including limit problems. Understanding how to handle these functions is crucial for evaluating limits. Importantly, the behavior of rational functions as variables approach certain values (like zero or infinity) is key to determining their limits. Rational functions make limit evaluation both interesting and challenging, as they incorporate polynomial degrees and coefficients which dictate their end behavior.

Infinity Limit

The concept of an infinity limit is all about understanding what happens to a function as the variable grows indefinitely large. In calculus, we often deal with limits where the variable heads towards infinity, like \( \lim_{n \to \infty} \frac{n}{2n+1} \). This indicates that we want to see the behavior of the function when the value of 'n' becomes very large. With infinity limits, the focus is usually on the dominant terms in the expression since they have the biggest impact when the variables grow. Evaluating these limits helps us determine the end behavior of functions and provides insights into how functions behave over long intervals or large inputs. Responding confidently to infinity limits requires a firm grasp of polynomial terms and their powers.

Leading Coefficients

Leading coefficients play a significant role in determining the behavior of rational functions at infinity. When you're examining a rational function for its limit at infinity, the terms with the highest powers in the numerator and the denominator are the most important. For example, in the expression \( \frac{n}{2n+1} \), both the numerator and the denominator have leading terms with the power of one ('n' in the numerator and '2n' in the denominator). The coefficients of these terms are key. Here, it's '1' for the numerator and '2' for the denominator, which simplifies the limit problem. Often, when the degrees of the numerator and denominator match, the limit at infinity simplifies to the ratio of these leading coefficients. Thus, understanding and identifying leading coefficients can simplify limit evaluation enormously.

Limit Evaluation Steps

Evaluating a limit involves systematic steps to simplify and solve the expression. When dealing with rational functions approaching infinity, start by identifying the type of limit. In this case, recognize that it’s an infinity limit of a rational function.Next, focus on simplifying the expression by dividing each term by the highest power of the variable present in the denominator. For \( \frac{n}{2n+1} \), divide each part by 'n', yielding \( \frac{1}{2 + \frac{1}{n}} \), which simplifies the analysis.As 'n' approaches infinity, terms like \( \frac{1}{n} \) approach 0. Incorporate this into your simplified expression to find the limit. Here, \( \frac{1}{2 + 0} \) simplifies to \( \frac{1}{2} \).Finally, confirm your calculation by revisiting each step and rationalizing the conclusion. Thoroughly follow these steps to accurately evaluate limits and deepen your understanding of calculus limits.