Problem 75 - Calculus for Biology and Medicine

Question

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(\frac{n^{2}+1}{n^{2}}\right) $$

Step-by-Step Solution

Answer

The limit is 1 as \(n\) approaches infinity.

Step 1: Analyze the Expression

We start by examining the expression inside the limit: \(\frac{n^2 + 1}{n^2}\). Notice that both the numerator and the denominator are polynomials in \(n\). As \(n\) approaches infinity, the highest degree terms will dominate the behavior of the expression.

Step 2: Simplify the Expression

To make the limit easier to evaluate, divide every term in the expression by \(n^2\) (the highest power in the denominator):\[\frac{n^2 + 1}{n^2} = \frac{n^2}{n^2} + \frac{1}{n^2} = 1 + \frac{1}{n^2}.\]

Step 3: Apply Limit Laws

Now, take the limit of the simplified expression:\[\lim_{n \to \infty} \left(1 + \frac{1}{n^2}\right).\]Using the limit law that \( \lim_{n \to \infty} (f(n) + g(n)) = \lim_{n \to \infty} f(n) + \lim_{n \to \infty} g(n) \), we find:\[\lim_{n \to \infty} 1 + \lim_{n \to \infty} \frac{1}{n^2}.\]

Step 4: Evaluate the Limits

Evaluate each part individually:- \(\lim_{n \to \infty} 1 = 1\) since the limit of a constant is the constant itself.- \(\lim_{n \to \infty} \frac{1}{n^2} = 0\) because as \(n\) approaches infinity, \(\frac{1}{n^2}\) approaches zero.

Step 5: Combine the Results

Combining these results, we have:\[\lim_{n \to \infty} \left(1 + \frac{1}{n^2}\right) = 1 + 0 = 1.\]

Key Concepts

Limit Laws

Limit laws are essential tools in calculus for evaluating limits in diverse expressions. They simplify the process of understanding the behavior of functions as the input approaches a particular value, often infinity. Among the most useful limit laws are the sum law and the constant law. The sum law states that the limit of the sum of two functions is the sum of their limits, assuming those limits exist. This principle was used in our solution to separate the terms in the expression:

• \(\lim_{n \to \infty} (1 + \frac{1}{n^2}) = \lim_{n \to \infty} 1 + \lim_{n \to \infty} \frac{1}{n^2}\)

The constant law confirms that the limit of a constant is just the constant itself, as shown when evaluating \(\lim_{n \to \infty} 1\). The ability to break down complex expressions into simpler parts using these laws supports efficient calculation and deepens understanding of function behavior at boundaries.

Polynomials

Polynomials are mathematical expressions consisting of variables raised to various powers. They play a significant role in limit problems because their structure becomes predictable at certain boundaries. In the provided exercise, both the numerator and the denominator are polynomials in terms of \(n\).The polynomial in both the numerator and the denominator is \(n^2\). As \(n\) becomes very large (approaching infinity), higher-order terms like \(n^2\) dominate the expression, while the lower-order terms (such as constant terms) become insignificant. This is due to the fact that in comparison to \(n^2\), constants and terms with lower powers of \(n\) shrink to zero.Simplifying polynomials by dividing through by the highest power of \(n\) in the denominator, as shown in the solution, simplifies evaluating the limit further:

• \(\frac{n^2 + 1}{n^2} = 1 + \frac{1}{n^2}\)

By focusing on the highest power, we can determine the end behavior of the polynomial expression efficiently.

Infinity

Infinity is a concept that doesn't represent a specific number but describes unbounded behavior. In calculus, dealing with infinity often involves understanding how expressions behave as variables grow without bounds. Evaluating limits as \(n\) approaches infinity means assessing the function's end behavior.When \(n\) tends towards infinity in expressions like \(\frac{1}{n^2}\), the value approaches zero. This is because as the denominator grows larger, the fraction becomes smaller. The concept of infinity helps us see that for extremely large values of \(n\), terms like these become negligible:

• The expression \(\lim_{n \to \infty} \frac{1}{n^2} = 0\) exemplifies this principle.

Infinity allows calculus to solve real-world problems by providing a lens to examine how mathematical expressions behave under extreme conditions, whether approaching vast sizes or minuscule zeros.