Problem 82 - Calculus for Biology and Medicine

Question

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty} \frac{n+3^{-n}}{n} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify the limit expression

We are given the expression \( \lim _{n \rightarrow \infty} \frac{n+3^{-n}}{n} \) and need to apply limit laws to evaluate it.

Step 2: Separate terms of the fraction

Rewrite the expression by separating the terms in the numerator: \( \frac{n+3^{-n}}{n} = \frac{n}{n} + \frac{3^{-n}}{n} \). This allows us to apply limit laws to each individual term.

Step 3: Apply limit laws to \(\frac{n}{n}\)

Since \(\frac{n}{n} = 1\), the limit of the first term as \(n\) approaches infinity is \( \lim _{n \rightarrow \infty} 1 = 1\).

Step 4: Simplify and evaluate \(\frac{3^{-n}}{n}\)

Notice that \(3^{-n}\) equals \(\left(\frac{1}{3}\right)^{n}\), which approaches 0 as \(n\) approaches infinity. Thus, \( \lim _{n \rightarrow \infty} \frac{3^{-n}}{n} = 0\) since \(3^{-n}\) decreases much faster than \(n\) increases.

Step 5: Combine results using limit addition law

Using the limit addition law, combine both parts: \( \lim _{n \rightarrow \infty} \left(\frac{n}{n} + \frac{3^{-n}}{n}\right) = \lim _{n \rightarrow \infty} 1 + \lim _{n \rightarrow \infty} 0 = 1 + 0 = 1\). Thus, the original limit is equal to 1.

Key Concepts

Infinity

Infinity is a concept that represents a quantity that is larger than any finite number. In mathematics, we often encounter infinity when evaluating limits, as variables approach infinitely large values or infinitely small values—often symbolized by the infinity symbol (∞). When dealing with limits, as we see in the problem, we evaluate what happens to an expression as our variable, in this case, \( n \), grows without bound.

• In this exercise, we are interested in what happens as \( n \) approaches infinity.
• As \( n \) becomes extremely large, certain terms like \( 3^{-n} \) diminish to nothing because of their exponential decay.

Understanding how terms behave as they approach infinity is key to utilizing limit laws effectively.

Separation of Terms

Separating terms is a useful technique in simplifying complex expressions, especially when dealing with limits. By breaking down a fraction into simpler components, we can individually evaluate each term. This is particularly helpful when applying limit laws.For the given expression \( \frac{n+3^{-n}}{n} \), separating the terms in the numerator allows us to handle them individually:

• Write the expression as \( \frac{n}{n} + \frac{3^{-n}}{n} \).

This separation:- Provides clarity in evaluation by exposing simple forms within the expression- Allows us to leverage limit laws, such as the limit of a sum being the sum of the limits

Limit Addition Law

The limit addition law is one of the fundamental limit laws that states: the limit of a sum is equal to the sum of the limits, given that the limits exist. In simpler terms, if you can find the limit of each part of an addition, then add these limits together.After separating the terms of our fraction, we apply the limit addition law:

• \( \lim _{n \rightarrow \infty} \left( \frac{n}{n} + \frac{3^{-n}}{n} \right) = \lim _{n \rightarrow \infty} \frac{n}{n} + \lim _{n \rightarrow \infty} \frac{3^{-n}}{n} \)
• Evaluating each term gives us \( 1 + 0 \)

Thus, the overall limit evaluates to 1. The addition law simplifies our ability to handle more complicated expressions by breaking them down into simpler, manageable components.

Simplifying Expressions

Simplifying mathematical expressions makes them easier to evaluate, especially in limit problems. By reducing expressions to their simplest form, we eliminate the noise and focus on the behavior of key components.In our exercise, we simplified the expression \( \frac{n+3^{-n}}{n} \) by:

• Splitting it into \( \frac{n}{n} \) and \( \frac{3^{-n}}{n} \).
• Recognizing that \( \frac{n}{n} \) simplifies to 1.
• Noticing that \( 3^{-n} \) becomes minuscule as \( n \) grows, effectively turning \( \frac{3^{-n}}{n} \) into 0.

Simplifying expressions is a critical step in evaluating limits effectively, allowing for precise and efficient calculation of complex mathematical relationships.