Problem 71 - Calculus for Biology and Medicine

Question

In Problems \(71-82\), use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) \(\lim _{n \rightarrow \infty}\left(\frac{1}{n}+\frac{1}{n^{2}}\right)\)

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Analyze the Expression

The given expression involved in taking the limit is \(\frac{1}{n} + \frac{1}{n^2}\). We need to consider the behavior of each term separately as \(n\) approaches infinity.

Step 2: Apply the Limit Law to First Term

Consider the first term \(\frac{1}{n}\). As \(n\) approaches infinity, \(\frac{1}{n}\) approaches zero because the denominator increases without bound, while the numerator remains constant.

Step 3: Apply the Limit Law to Second Term

Now, consider the second term \(\frac{1}{n^2}\). Similarly, as \(n\) approaches infinity, \(\frac{1}{n^2}\) approaches zero since, again, the denominator grows significantly faster than the numerator.

Step 4: Combine the Limits

Using the limit law that allows the sum of limits, we can combine the limits: \(\lim_{n \to \infty} \left( \frac{1}{n} + \frac{1}{n^2} \right) = \lim_{n \to \infty} \frac{1}{n} + \lim_{n \to \infty} \frac{1}{n^2}\). Both of these individual limits equal zero.

Step 5: Conclude the Result

The sum of two zero limits is also zero. Thus, \(\lim_{n \to \infty} \left( \frac{1}{n} + \frac{1}{n^2} \right) = 0\). Therefore, \(a = 0\).

Key Concepts

Understanding Infinite Limits

When we talk about infinite limits, we refer to the behavior of a function as its input approaches a certain point, often infinity. In the original exercise, we used this concept to examine \(\lim_{n \rightarrow \infty} \left( \frac{1}{n} + \frac{1}{n^2} \right)\).
This tells us how the expression behaves as \(n\) becomes very large. The key steps include:

• Analyze Each Term: Look at \(\frac{1}{n}\) and \(\frac{1}{n^2}\) individually. As \(n\) increases, each component's value approaches zero.
• Apply Limit Laws: The sum becomes zero since both individual limits are zero.

The process underscores the utility of limit laws when dealing with infinite sums, particularly when multiple expressions are involved.

Grasping Sequence Convergence

Sequence convergence is another way to explore behavior as a sequence progresses. A sequence like \(a_n = \frac{1}{n} + \frac{1}{n^2}\) is said to converge if, as \(n\) grows larger, its terms approach a specific value. In this case, that value is zero.Here's how it works:

• Identify the Limit: Both \(\frac{1}{n}\) and \(\frac{1}{n^2}\) approach zero as \(n\) heads toward infinity.
• Combine and Conclude: Since both components approach zero, the sequence \(a_n\) converges to zero.

Essentially, you follow each term's path and conclude the sum’s behavior. This realization is crucial for understanding how individual components impact overall sequence convergence.

The Role of Calculus Series in Limits

Calculus series are sequences of numbers added together based on a limit process. In our original problem, we had a series of terms \(\frac{1}{n} + \frac{1}{n^2}\) converging to a final limit as \(n\) becomes infinitely large.Consider these points:

• Series Representation: A series can be a simple sum like \(\frac{1}{n} + \frac{1}{n^2}\) or can include more complex terms.
• Summing the Limits: By analyzing each term, we identified the series converges to zero following fundamental limit laws.

Understanding calculus series is key to solving complex limit problems, allowing us to piece together individual behaviors for comprehensive solutions.

Understanding Asymptotic Behavior

Asymptotic behavior describes how functions behave at infinity or near certain points. Asymptotic analysis in our limit problem reveals how the expression becomes negligible for large \(n\). Here's what we mean:

• Asymptotes at Infinity: For \(\frac{1}{n}\) and \(\frac{1}{n^2}\), as \(n\) gets extremely large, these terms trend towards zero due to their higher power denominators.
• Practical Outcomes: Predicts the convergence point as \(n\) grows without end, illustrating why these terms diminish in influence.

Asymptotic behavior helps anticipate overall patterns in functions or sequences, particularly when handling such limits.