Problem 10 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

Bestimmen Sie die Grenzwerte (1) \(\lim _{n \rightarrow \infty} \frac{6 n^{2}+5 n}{4 n^{2}+n+1}\), (2) \(\frac{1 i m}{n \rightarrow \infty} \frac{8 n^{5}+9 n^{3}+7}{n^{6}+3 n}\); (3) \(\lim _{n \rightarrow \infty}\left(\frac{8 n}{3 n+1}\right)^{3}\), (4) \(\lim _{n \rightarrow \infty}\left(\frac{2 n(n+1)}{n+2}-\frac{2 n^{3}}{n^{2}+2}\right)\) (5) \(\lim (\sqrt{n+1}-\sqrt{n})\).

Step-by-Step Solution

Answer

(1) \(\frac{3}{2}\), (2) 0, (3) \(\frac{512}{27}\), (4) 0, (5) 0.

Step 1: Simplify the Expression for (1)

Consider the dominant terms in the numerator and denominator of \(\frac{6n^2 + 5n}{4n^2 + n + 1}\). Divide both the numerator and the denominator by \(n^2\), the highest power in the polynomial.\[\lim_{n \to \infty} \frac{6n^2/n^2 + 5n/n^2}{4n^2/n^2 + n/n^2 + 1/n^2} = \lim_{n \to \infty} \frac{6 + 5/n}{4 + 1/n + 1/n^2}\]As \(n\) approaches infinity, the terms \(5/n, 1/n,\) and \(1/n^2\) tend to zero. Thus, we have:\[\lim_{n \to \infty} \frac{6}{4} = \frac{3}{2}\]

Step 2: Simplify the Expression for (2)

Analyze the expression \(\frac{8n^5 + 9n^3 + 7}{n^6 + 3n}\). The highest power is \(n^6\) in the denominator. Divide both the numerator and the denominator by \(n^6\).\[\lim_{n \to \infty} \frac{8n^5/n^6 + 9n^3/n^6 + 7/n^6}{n^6/n^6 + 3n/n^6} = \lim_{n \to \infty} \frac{8/n + 9/n^3 + 7/n^6}{1 + 3/n^5}\]As \(n\) approaches infinity, all terms in the numerator tend to zero; hence:\[\lim_{n \to \infty} \frac{0}{1} = 0\]

Step 3: Exponentiate the Simplified Fraction for (3)

Consider \(\left(\frac{8n}{3n+1}\right)^3\) and simplify inside the parenthesis by dividing the numerator and denominator by \(n\).\[\lim_{n \to \infty} \left(\frac{8n/n}{3n/n + 1/n}\right)^3 = \lim_{n \to \infty} \left(\frac{8}{3 + 1/n}\right)^3\]As \(n\) approaches infinity, \(1/n\) tends to zero:\[\lim_{n \to \infty} \left(\frac{8}{3}\right)^3 = \left(\frac{8}{3}\right)^3 = \frac{512}{27}\]

Step 4: Evaluate the Expression in (4)

Let's simplify \(\frac{2n(n+1)}{n+2} - \frac{2n^3}{n^2+2}\). Simplify each term separately by dividing by the highest power in their respective denominators.For \(\frac{2n(n+1)}{n+2}\):\[\lim_{n \to \infty} \frac{2n^2 + 2n}{n+2} \approx \frac{2n^2}{n} = 2n\]For \(\frac{2n^3}{n^2+2}\):\[\lim_{n \to \infty} \frac{2n^3}{n^2} = 2n\]Consequently, the overall limit is:\[\lim_{n \to \infty} (2n - 2n) = 0\]

Step 5: Analyze the Square Roots in (5)

Consider \(\sqrt{n+1} - \sqrt{n}\). Rationalize by multiplying the numerator and the denominator by the conjugate \(\sqrt{n+1} + \sqrt{n}\).\[\lim_{n \to \infty} \frac{(\sqrt{n+1} - \sqrt{n})(\sqrt{n+1} + \sqrt{n})}{\sqrt{n+1} + \sqrt{n}} = \lim_{n \to \infty} \frac{1}{\sqrt{n+1} + \sqrt{n}}\]As \(n\) approaches infinity:\[\lim_{n \to \infty} \frac{1}{2\sqrt{n}} = 0\]

Key Concepts

Polynomdivision

To handle limits of expressions involving polynomials, polynomial division is a crucial tool. It helps simplify the fraction by dividing the terms of the highest power, usually leading the behavior as the variable goes to infinity. For example, in the problem where we simplify \( \frac{6n^2 + 5n}{4n^2 + n + 1} \), polynomial division involves dividing both the numerator and the denominator by \( n^2 \), the highest power present in the denominator. This simplifies the expression so that as \( n \to \infty \), the lower power terms diminish and we observe the behavior of simpler constants. This method helps us isolate the leading coefficient ratio, which in turn gives us the limit.

Konjugierte Rationalisierung

Conjugate rationalization is a technique useful when dealing with limits of expressions involving square roots. This method involves multiplying the numerator and the denominator by the conjugate of the numerator. For instance, in the expression \( \sqrt{n+1} - \sqrt{n} \), we rationalize by multiplying and dividing by its conjugate \( \sqrt{n+1} + \sqrt{n} \). This transforms the expression to one where subtraction of squares occurs, allowing the elimination of the square roots. As we simplify, it becomes easier to compute the limit, especially as \( n \to \infty \), since the resultant expression tends to a simpler form that highlights the dominance of terms as \( n \) grows large.

Asymptotisches Verhalten

Understanding the asymptotic behavior of expressions is essential when evaluating limits as a variable approaches infinity. Consider the example \( \left(\frac{8n}{3n+1}\right)^3 \). Here, asymptotic behavior is analyzed by dividing the numerator and denominator by \( n \), the common factor, which reveals that as \( n \to \infty \), the term \( \frac{1}{n} \) becomes negligible. Thus, the primary behavior of the expression is dominated by the leading coefficients, resulting in a simplified computation of the limit. This behavior reflects the approach to identify and understand the main trend or growth of an expression as variables become very large.

Mathematische Analyse

Mathematical analysis provides us with powerful techniques to evaluate limits precisely, including those of complex expressions. Through the use of detailed algebraic manipulation, like division and rationalization, we transform intricate expressions into forms that are easier to evaluate. For instance, when evaluating \( \frac{2n(n+1)}{n+2} - \frac{2n^3}{n^2+2} \), we separately simplify each fraction by considering the highest degree terms, highlighting the importance of analyzing the most influential parts of the expression. Combining techniques allows for clearer visualization of the function's behavior, ultimately making it simpler to determine the limit as \( n \to \infty \). Mathematical analysis thus encompasses various strategies and manipulations to gain insight into function behaviors seamlessly.