Problem 76 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is 3.

Step 1: Identify the Function Inside the Limit

The function we are examining is \( \frac{3n^2 - 5}{n^2} \). This rational function involves polynomials in the numerator and the denominator.

Step 2: Simplify the Rational Expression

Divide every term of the numerator and the denominator by \( n^2 \) to simplify the expression:\[\frac{3n^2 - 5}{n^2} = \frac{3n^2}{n^2} - \frac{5}{n^2} = 3 - \frac{5}{n^2}\].

Step 3: Apply Limit to Each Term Individually

Use the limit laws to evaluate the limit of each term separately. We have:\[\lim_{n \to \infty} \left(3 - \frac{5}{n^2}\right) = \lim_{n \to \infty} 3 - \lim_{n \to \infty} \frac{5}{n^2}\].

Step 4: Evaluate the Limits

The limit \( \lim_{n \to \infty} 3 \) is simply 3 because it is a constant. For \( \lim_{n \to \infty} \frac{5}{n^2} \), since \( \frac{5}{n^2} \) approaches 0 as \( n \) approaches infinity, the limit is 0.

Step 5: Combine the Results

Combine the results from the previous step:\[3 - 0 = 3\]. Therefore, the overall limit is 3.

Key Concepts

Rational Functions

A rational function is a type of function that can be expressed as the ratio of two polynomials. This means it takes the form \( \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials, and \( q(x) \) is not zero. In the given problem, the rational function is \( \frac{3n^2 - 5}{n^2} \).

Rational functions are very flexible. They can model various behaviors depending on the degrees and coefficients of the polynomials involved. For example:

• If the degree of the polynomial in the numerator is higher, the function tends to infinity.
• If the degree is the same, the function tends towards the ratio of the leading coefficients.
• If the degree of the denominator is higher, the function approaches zero.

Understanding these characteristics allows us to predict the behavior of rational functions as their variables grow large, which is central to solving limits involving these functions. By simplifying them by dividing every term by the highest degree in the denominator, we can easily apply limit laws.

Polynomials

Polynomials are fundamental mathematical expressions consisting of variables and coefficients. They are comprised of terms of the form \( ax^n \), where \( n \) is a non-negative integer and \( a \) is a coefficient. The polynomial in the numerator of the original exercise is \( 3n^2 - 5 \), while the denominator polynomial is simply \( n^2 \).

The degree of a polynomial is determined by the highest power of the variable present in it. In the example at hand, the degree of both the polynomial in the numerator and the denominator is 2. This is pivotal because:

• When calculating limits for rational functions, the degrees of these polynomials help us determine the behavior of the function as \( n \) approaches infinity.
• Since the degrees in both the numerator and denominator are equal, simplifying this polynomial means that the leading coefficients (3 from \( 3n^2 \) and 1 from \( n^2 \)) become crucial in evaluating the limit.
• This property makes it easier to analyze limits and apply limit laws by focusing on dominant terms.

Polynomials help define these behaviors that lead to easy simplification steps, making the calculation of limits more straightforward.

Infinite Limits

Infinite limits describe the behavior of functions as the input values grow indefinitely large. In the current exercise, we are interested in determining the behavior of \( \lim_{n \to \infty} \frac{3n^2 - 5}{n^2} \). Dealing with infinity in limits involves understanding how each part of a function behaves as the variable \( n \) grows without bound.

To solve for infinite limits, we employ:

• **Simplification**: We simplify the expression \( \frac{3n^2 - 5}{n^2} \) by dividing each term by \( n^2 \), which represents the highest degree in the denominator. This yields \( 3 - \frac{5}{n^2} \).
• **Limit Laws**: Apply these to each term separately. For the constant 3, the limit remains unchanged, while \( \frac{5}{n^2} \) approaches 0 since the denominator \( n^2 \) becomes very large.
• **Combining Results**: Merge these into a single outcome to conclude that the limit equals 3.

Understanding infinite limits is essential for mastering calculus. They help indicate how functions behave at specific boundaries, like infinity, and are critical to analyzing functions that either grow without bound or stabilize at a particular value.