Problem 84 - Calculus for Biology and Medicine

Question

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(\frac{3 n^{2}-5}{n}\right) $$

Step-by-Step Solution

Answer

The limit is infinite, so it does not exist.

Step 1: Identify the Function

The given function is \( f(n) = \frac{3n^2 - 5}{n} \). We need to find \( \lim_{n \to \infty} f(n) \) using limit laws.

Step 2: Simplify the Expression

Divide both the numerator and the denominator by \( n \): \( \frac{3n^2/n - 5/n}{n/n} = \frac{3n - \frac{5}{n}}{1} \).

Step 3: Apply the Limit Law

Use the limit law that states \( \lim_{n \to \infty} \left( rac{a_n}{b_n} \right) = \frac{\lim_{n \to \infty} a_n}{\lim_{n \to \infty} b_n} \) if \( \lim_{n \to \infty} b_n eq 0 \). Here, \( \lim_{n \to \infty} \frac{5}{n} = 0 \) and \( \lim_{n \to \infty} n = \infty \).

Step 4: Simplify Further Using Limit Properties

Since \( \frac{5}{n} \) approaches 0 as \( n \) approaches infinity, the expression simplifies to \( \lim_{n \to \infty} (3n - 0) = \lim_{n \to \infty} 3n \).

Step 5: Conclude the Limit Computation

Since \( \lim_{n \to \infty} 3n \to \infty \), the function \( f(n) = \frac{3n^2 - 5}{n} \) grows without bound as \( n \to \infty \). Therefore, the limit does not exist (is infinite).

Key Concepts

Infinite Limits

When a variable in a function approaches a particular value, and the output of the function increases indefinitely without settling at a certain number, we are dealing with infinite limits. These occur when functions grow larger and larger in positive or negative directions as the variable approaches some value, often infinity. In the context of this exercise, as the input value \( n \) approaches infinity, the expression \( \frac{3n^2 - 5}{n} \) behaves in such a manner, indicating the presence of an infinite limit.

Key points to understand infinite limits include:

• Not all limits yield finite values. Sometimes, the result is that the function grows without bound, either positively or negatively.
• This can often be observed in rational functions (those that have a variable in the denominator) and polynomial functions as their degree dictates their rate of growth.
• Knowing how to identify when a limit becomes infinite helps in understanding the behavior of functions as they extend towards extremely large or small input values.

In this exercise, infinity in the limit indicates that \( f(n) \) does not approach a specific real number but instead increases forever as \( n \) becomes very large.

Rational Functions

A rational function is any function that can be expressed as the ratio of two polynomials. The general form of a rational function is \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. They are particularly important in calculus as they often exhibit many interesting behaviors, such as asymptotes and discontinuities.

In the given problem, \( f(n) = \frac{3n^2 - 5}{n} \) is a rational function because it involves the polynomial \( 3n^2 - 5 \) in the numerator and the simple polynomial \( n \) in the denominator. Understanding rational functions is critical because:

• They can have horizontal or vertical asymptotes, which describe behavior as values approach infinity or particular critical points.
• They may exhibit removable discontinuities or holes, depending on the factors of \( P(x) \) and \( Q(x) \).
• The degree of the polynomials involved significantly affects the function's end behavior as \( x \) approaches infinity.

For \( \frac{3n^2 - 5}{n} \), simplifying the degrees helps identify that the leading term in the numerator dictates the outcome as \( n \) increases without bound, reflecting an infinite limit.

Simplification of Expressions

Simplifying mathematical expressions is an essential step that makes finding limits easier, especially when dealing with rational functions. Simplification reduces complexity and helps in identifying the dominant terms that really affect the limit.

To simplify \( \frac{3n^2 - 5}{n} \), divide every term by \( n \), which results in \( 3n - \frac{5}{n} \). This process is invaluable:

• By simplifying, the dominant terms become clear. For infinite limits, these are the terms with the highest powers of \( n \).
• It helps in applying limit laws more effectively, as simplified expressions are easier to evaluate and understand.
• Through simplification, it becomes evident which terms contribute to the behavior as values approach infinity or other asymptotic points.

Here, the simplification shows that as \( n \) tends to infinity, \( \frac{5}{n} \) tends towards zero, leaving \( 3n \) as the primary influencer of the limit. As a result, the function's growth to infinity becomes clearly visible. This simplification step thus lays the groundwork for concluding that the limit of the function as \( n \) approaches infinity is indeed infinite.