Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the terms in the expression

The expression given is \( \frac{n}{2n+1} \). The numerator is \( n \) and the denominator is \( 2n + 1 \). As \( n \) approaches infinity, both the numerator and denominator grow.

Step 2: Simplify the fraction by dividing by n

To simplify the expression, divide both the numerator and the denominator by \( n \):\[\frac{n}{2n+1} = \frac{n/n}{(2n/n) + (1/n)} = \frac{1}{2 + \frac{1}{n}}\].

Step 3: Evaluate the limit as n approaches infinity

As \( n \rightarrow \infty \), the term \( \frac{1}{n} \rightarrow 0 \). Therefore, the expression simplifies to:\[\lim_{n \rightarrow \infty} \frac{1}{2 + \frac{1}{n}} = \frac{1}{2 + 0} = \frac{1}{2}\].

Step 4: State the final result

The limit of the sequence as \( n \) approaches infinity is \( \frac{1}{2} \).

Key Concepts

Infinity in Calculus

Infinity in calculus refers to the behavior of functions as inputs become infinitely large or small. It's a way to describe trends where values move closer and closer to a boundary without necessarily reaching it. In this particular exercise, we are dealing with the limit as \( n \) approaches infinity. This gives us insight into the behavior of the expression \( \frac{n}{2n+1} \) as \( n \) gets increasingly large. Understanding infinity is crucial in calculus because it helps us comprehend the long-term trends and asymptotic behaviors of functions.
Different scenarios involve infinity in calculus, including the calculation of limits involving infinite sequences or series. Grasping the concept of infinity can help students understand how algorithms operate in extreme cases and how they can describe real-world phenomena like spreading networks or financial models.

Fraction Simplification

Fraction simplification is an important skill in mathematics, especially when dealing with limits and calculus. It allows us to transform complex expressions into simpler forms that are easier to evaluate. In this exercise, we simplified the fraction \( \frac{n}{2n+1} \) by dividing each term by \( n \).
This step is crucial because it makes the limit evaluation manageable:

• The numerator \( n \) divided by \( n \) gives 1.
• In the denominator, \( 2n \) divided by \( n \) becomes 2, and the constant \( 1 \) divided by \( n \), \( \frac{1}{n} \), becomes a term that diminishes as \( n \) grows.

Knowing how to simplify fractions by eliminating common factors or reducing terms can make complex calculus problems much easier to solve.

Evaluating Limits

Evaluating limits is a foundational technique in calculus, allowing us to find the value that a function approaches as the input reaches some point. To determine the limit of \( \frac{n}{2n+1} \) as \( n \to \infty \), we approached it step by step:

• Firstly, simplify the expression by dividing each component by \( n \).
• Next, identify terms that become negligible as \( n \) becomes infinitely large, such as \( \frac{1}{n} \) which tends to zero.

This reduction leads to the simplified expression \( \frac{1}{2} \), as the negligible terms drop out.
Understanding the subtle behavior of functions as they grow helps in many areas of calculus, such as understanding continuity, optimizing processes, and solving differential equations.

Step-by-Step Solution

Following a step-by-step solution is a robust method to tackle complex calculus problems. It breaks down complicated expressions into easily digestible parts, ensuring you don't miss out on critical steps. Here's how we applied this strategy:

• Identify terms: We first recognized the main components of the expression \( \frac{n}{2n+1} \), identifying them as growing with \( n \).
• Simplify fractions: By dividing by \( n \), this step helped simplify the terms involved, cleanly setting up the next steps.
• Evaluate the limit: We examined the behavior as \( n \to \infty \), focusing on reducing it to a simpler expression, \( \frac{1}{2} \).
• State the result: Finally, we articulated our findings clearly, explaining the reasoning behind each move.

Adopting a step-by-step approach not only enhances understanding but also builds a habit of organized problem-solving, which proves immensely beneficial for tackling diverse calculus challenges.