Question

Find \(\lim _{n \rightarrow \infty} s_{n}\). Justify your answers from Definition 4.1.1. (a) \(s_{n}=\frac{n}{2 n+\sqrt{n+1}}\) (b) \(s_{n}=\frac{n^{2}+2 n+2}{n^{2}+n}\) (c) \(s_{n}=\frac{\sin n}{\sqrt{n}}\) (d) \(s_{n}=\sqrt{n^{2}+n}-n\)

Short answer

Question: Determine the limit of the following sequences as n approaches infinity and justify your answers using Definition 4.1.1: (a) \(s_{n}=\frac{n}{2 n+\sqrt{n+1}}\) (b) \(s_{n}=\frac{n^{2}+2 n+2}{n^{2}+n}\) (c) \(s_{n}=\frac{\sin n}{\sqrt{n}}\) (d) \(s_{n}=\sqrt{n^{2}+n}-n\) Answer: (a) The limit of the sequence \(s_n\) as n approaches infinity is \(\frac{1}{2}\). (b) The limit of the sequence \(s_n\) as n approaches infinity is \(1\). (c) The limit of the sequence \(s_n\) as n approaches infinity is \(0\). (d) The limit of the sequence \(s_n\) as n approaches infinity is \(\frac{1}{2}\).

Step-by-step solution

Find the limit

By dividing both numerator and denominator of \(s_n\) by \(n\), we have: $$s_n = \frac{1}{2 + \frac{\sqrt{n+1}}{n}}$$ As \(n \rightarrow \infty\), \(\frac{\sqrt{n+1}}{n} \rightarrow 0\). Thus, the limit of \(s_n\) can be found as follows: $$ \lim _{n \rightarrow \infty} s_{n} = \frac{1}{2} . $$ (b) \(s_{n}=\frac{n^{2}+2 n+2}{n^{2}+n}\)

Find the limit

Divide both numerator and denominator of \(s_n\) by \(n^2\), we have: $$s_n = \frac{1+\frac{2}{n}+\frac{2}{n^2}}{1 + \frac{1}{n}}$$ As \(n \rightarrow \infty\), \(\frac{2}{n} \rightarrow 0\) and \(\frac{2}{n^2} \rightarrow 0\), \(\frac{1}{n} \rightarrow 0\). Thus, the limit of \(s_n\) can be found as follows: $$ \lim _{n \rightarrow \infty} s_{n} = 1 . $$ (c) \(s_{n}=\frac{\sin n}{\sqrt{n}}\)

Find the limit

Before invoking the definition, note that for all \(n\), $$ -1 \le \sin n \le 1 .$$ Multiply this inequality by \(\frac{1}{\sqrt{n}}\): $$ -\frac{1}{\sqrt{n}} \le s_n \le \frac{1}{\sqrt{n}} .$$ As \(n \rightarrow \infty\), \(\frac{1}{\sqrt{n}} \rightarrow 0\). By squeezing principle, the limit of \(s_n\) can be found as follows: $$ \lim_{n \rightarrow \infty} s_{n} = 0 . $$ (d) \(s_{n}=\sqrt{n^{2}+n}-n\)

Find the limit

Multiply and divide by the conjugate of the expression inside the square root to simplify the expression: $$ s_n = \sqrt{n^{2}+n}-n \cdot \frac{\sqrt{n^{2}+n}+n}{\sqrt{n^{2}+n}+n} = \frac{n}{\sqrt{n^{2}+n}+n} .$$ Now, divide both numerator and denominator of \(s_n\) by \(n\): $$s_n = \frac{1}{\frac{\sqrt{n^2+n}}{n}+1} .$$ As \(n \rightarrow \infty\), \(\frac{\sqrt{n^2+n}}{n} \rightarrow 1\). Thus, the limit of \(s_n\) can be found as follows: $$ \lim_{n \rightarrow \infty} s_{n} = \frac{1}{2} .$$

Key concepts

Definition of a Limit

In mathematics, the limit of a sequence is a fundamental concept that helps us understand the behavior of sequences as they progress towards infinity. More specifically, we often want to know if a sequence of numbers approaches a particular value as the terms increase. The formal definition of a limit says that for any given small positive number \( \varepsilon \), no matter how tiny, we can find a term in the sequence beyond which all subsequent terms are within \( \varepsilon \) of some real number \( L \), which is the limit of the sequence.

In simpler terms, if we can make the terms of a sequence as close as we want to a specific value by just going far enough out, then the sequence has a limit equal to that value. This understanding is crucial for various calculations in calculus and real analysis and guides us in solving limits of functions and series.

Squeeze Theorem

The Squeeze Theorem, also known as the sandwich theorem, is a useful tool in finding the limit of a sequence, especially when dealing directly with the sequence is complex or difficult. It involves three sequences: \( a_n \), \( b_n \), and \( c_n \). According to the theorem, if \( a_n \leq b_n \leq c_n \) for all \( n \) large enough, and both \( \lim_{n \to \infty} a_n = L \) and \( \lim_{n \to \infty} c_n = L \) exist, then \( \lim_{n \to \infty} b_n = L \) as well.

Here's how it applies: imagine two sequences converging to the same limit, and there's a third sequence sandwiched between them. If the outer sequences squeeze their way towards a particular value \( L \), the middle sequence also converges to \( L \). In practical use, this means that if you can bound a complex sequence between two simpler ones whose limits are known, the challenging sequence shares the same limit.

Conjugate Multiplication

Conjugate multiplication is a powerful trick often used in algebra and calculus to simplify expressions involving radicals. It involves multiplying by a conjugate to rationalize denominators or to get rid of irrational parts in expressions. The conjugate of an expression \( a + b \sqrt{c} \) is \( a - b \sqrt{c} \), and multiplying by the conjugate results in a difference of squares, which simplifies the expression.

For instance, in the expression \( \sqrt{n^2 + n} - n \), multiplying by its conjugate \( \sqrt{n^2 + n} + n \) results in eliminating the square root in the numerator, making it easier to handle. This technique is particularly helpful in finding limits involving roots, as seen in some limit problems, where direct computation isn't straightforward. By simplifying complex expressions, it opens a path to identify the behavior of sequences or functions.

Mathematical Justification

Mathematical justification is the process of providing a solid reasoning for each step taken during problem-solving, especially in exercises involving complex calculus concepts like limits. It's not enough to state the result; understanding how and why a result is reached solidifies comprehension and validates the correctness of a solution.

Justification involves referring to known definitions, theorems, or lemmas that support the transitions or transformations applied to mathematical expressions during computations. For limits, this often means citing definitions such as the definition of a limit, or theorems like the Squeeze Theorem, to explain why a given sequence converges to a particular limit. Ensuring every step aligns logically with previous mathematical principles or proofs provides credibility and clarity to the solution, ultimately enhancing learning and reinforcing knowledge.