Problem 73 - Calculus for Biology and Medicine

Question

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(\frac{n+1}{n}\right) $$

Step-by-Step Solution

Answer

The limit is 1 as \( n \to \infty \).

Step 1: Simplify the expression

First, simplify \( \frac{n+1}{n} \). Rewrite the fraction by dividing each term in the numerator by the denominator \( n \): \[ \frac{n+1}{n} = \frac{n}{n} + \frac{1}{n} = 1 + \frac{1}{n}. \]

Step 2: Apply the limit laws

Apply the limit laws by considering each term separately. We have: \[ \lim_{n \to \infty} \left(1 + \frac{1}{n}\right) = \lim_{n \to \infty} 1 + \lim_{n \to \infty} \frac{1}{n}. \]

Step 3: Evaluate each limit

Evaluate each part of the sum. The limit of a constant is the constant itself, so \( \lim_{n \to \infty} 1 = 1 \). For the second term, as \( n \to \infty \), \( \frac{1}{n} \to 0 \). So, we have: \[ \lim_{n \to \infty} \frac{1}{n} = 0. \]

Step 4: Combine the results

Combine the results from Step 3: \[ \lim_{n \to \infty} \left(1 + \frac{1}{n}\right) = 1 + 0 = 1. \] Therefore, \( a = 1 \).

Key Concepts

Limits in Calculus

Calculus is the study of change and limits are fundamental to understanding this subject. A limit in calculus helps us grasp the behavior of a function as it approaches a specific point. It's like zooming in on a curve to see exactly where it heads or stops.

When we talk about a limit, we're often interested in what happens as our variable 'n' gets very large or very small. If we use the example from the exercise, \[ \lim_{n \rightarrow \infty} \left( \frac{n+1}{n} \right), \]we are observing how the expression behaves as \( n \) approaches infinity.

Here, the limit laws come into play. These laws are rules that help us simplify complicated limits by breaking them down into simpler parts. For example, we can separate constants from the variables, apply limits to each, and then combine results to find the overall limit. Mastery of limit laws is crucial in calculus as they form the backbone of more complex calculations.

Infinite Limits

Infinite limits occur when a function's value grows without bound as our variable approaches a certain point. Specifically, it means checking if the function heads towards infinity or a well-defined value when the variable goes to infinity.

In the exercise, we evaluate \[ \lim _{n \to \infty} \left( \frac{n+1}{n} \right). \]As \( n \) grows, the term \( \frac{1}{n} \) gets closer and closer to zero. This becomes clear when we simplify the expression.

Understanding infinite limits involves getting comfortable with the behavior of terms like \( \frac{1}{n} \). As \( n \) becomes infinitely large, the effect of adding \( 1 \) becomes increasingly negligible. Therefore, even when values tend to infinity, the function might stabilize to a particular value, as we saw in the solution (\( a = 1 \)).

Recognizing when a function stabilizes or heads to infinity is essential for comprehending its long-term behavior, an integral part of calculus and mathematical analysis.

Simplifying Expressions

To master limits, one must also be adept at simplifying expressions. This process helps break down complex terms into simpler, more manageable parts, making it easy to apply limit laws.

In the given exercise, the expression \( \frac{n+1}{n} \)was simplified by separating the terms in the numerator and dividing each by \( n \). This results in:\[ \frac{n}{n} + \frac{1}{n} = 1 + \frac{1}{n}. \]

Simplifying helps us to see that as \( n \rightarrow \infty, \frac{1}{n} \rightarrow 0. \)This simplification shows that the expression approaches 1 as \( n \) grows larger, which is why the limit of the entire expression was determined to be 1.

By regularly practicing the skill of simplifying expressions, you develop intuition about how different components behave in limit scenarios. It’s like decluttering a messy room; it clears the path for a much more straightforward understanding of how the subject works and aids in finding limits quickly and accurately.