Problem 25 - Essential Calculus

Question

In Exercises \(25-28,\) find the limit (if possible) of the sequence. \(a_{n}=\frac{5 n^{2}}{n^{2}+2}\)

Step-by-Step Solution

Answer

The limit of the sequence \(a_{n}=\frac{5 n^{2}}{n^{2}+2}\) as \(n\) goes to infinity is 5.

Step 1: Recognize the form

The sequence is a rational function of the form \(\frac{P(n)}{Q(n)}\) where \(P(n) = 5n^2\) and \(Q(n)=n^2+2\).

Step 2: Simplify the sequence

The sequence can be simplified by dividing every term in the numerator and the denominator by \(n^2\). Doing so results in \(a_{n}=\frac{5}{\frac{n^{2}+2}{n^{2}}}\). This simplifies further to \(a_{n}=\frac{5}{1+\frac{2}{n^2}}\).

Step 3: Determine the limit

Determining the limit as \(n\) goes to infinity, the term \(\frac{2}{n^2}\) goes to zero since the denominator is going to infinity. Thus, the whole sequence tends towards \(\frac{5}{1+0}=5\).

Key Concepts

Rational Function

In mathematics, a rational function is a function that can be expressed as the ratio of two polynomials. It is in the form of \( R(n) = \frac{P(n)}{Q(n)} \) where \( P(n) \) and \( Q(n) \) are polynomials in \( n \) with \( Q(n) \) not equal to zero. In the given exercise, the sequence \( a_{n} = \frac{5n^2}{n^2+2} \) is a rational function where the numerator \( P(n) = 5n^2 \) and the denominator \( Q(n) = n^2 + 2 \).

Understanding rational functions is vital since they appear in a variety of mathematical and applied contexts, from solving real-world problems to analyzing complex theories. These functions often describe rates or ratios that occur naturally in the sciences and finance, making them invaluable tools for many fields.

Simplifying Sequences

Simplifying sequences, especially those involving rational functions, requires algebraic manipulation to make them easier to evaluate or understand. In the \( a_{n} = \frac{5n^2}{n^2+2} \) example, simplification involves dividing the numerator and denominator by \( n^2 \) because it's the highest power of \( n \) common to both terms. This step yields \( a_{n} = \frac{5}{1 + \frac{2}{n^2}} \).

Simplification is a crucial step in mathematical problem-solving. It can help reveal patterns or properties that are not immediately obvious. When dealing with sequences, simplification often leads to easier calculations of terms and, as we'll see in the next section, understanding of behavior as the sequence progresses towards its limit.

Practical Tip for Simplification

To simplify a sequence efficiently, look for common factors, highest powers, and patterns that can be factored out or cancelled. These techniques decrease complexity and lay a clearer path toward finding limits or other characteristics of the sequence.

Limits at Infinity

The concept of \( \text{limits at infinity} \) is a cornerstone in calculus. It involves finding the value that a sequence or function approaches as the variable grows without bound. So, when considering sequences like \( a_{n} \) from our exercise, the question is: what value does \( a_{n} \) approach as \( n \) goes to infinity?

In the step-by-step solution, we determined that as \( n \) becomes very large, the term \( \frac{2}{n^2} \) gets closer and closer to zero because the denominator (\( n^2 \) grows faster than the numera`tor (\( 2 \)). This observation simplifies the sequence to \( \frac{5}{1 + 0} \) which equals to \( 5 \). Thus, the limit of the sequence \( a_{n} \) is \( 5 \).

Understanding limits at infinity is critical for predicting long-term behavior of sequences and functions. This concept is often used in calculus to describe asymptotes, optimize functions, and even in probability theory. It is also essential for the analysis of series convergence or divergence, which has implications across both pure and applied mathematics.