Question

Prove: If \(\lim _{n \rightarrow \infty} s_{n}=s\) and \(\lim _{n \rightarrow \infty} t_{n}=t,\) then $$ \lim _{n \rightarrow \infty} \frac{s_{n}}{t_{n}}=\frac{s}{t} $$ if \(s / t\) is defined in the extended reals and \(t \neq 0 .\) Give an example where \(s / t\) is defined in the extended plane, but (A) does not hold.

Short answer

Question: Prove that if the sequences $(s_n)$ and $(t_n)$ have limits $s$ and $t$ respectively, and $\frac{s}{t}$ is defined in the extended reals, then the limit of the sequence $\frac{s_n}{t_n}$ is equal to $\frac{s}{t}$. Provide a counterexample where (A) does not hold. Answer: We proved that if the limits $s$ and $t$ exist and $\frac{s}{t}$ is defined in the extended reals, then the limit of the sequence $\frac{s_n}{t_n}$ is equal to $\frac{s}{t}$. A counterexample is the sequences $s_n = n$ and $t_n = 2n$, where the limits are both infinite, making $\frac{s}{t}$ undefined in the extended reals. However, the limit of the sequence $\frac{s_n}{t_n}$ does converge to $\frac{1}{2}$, showing that (A) does not hold in this case.

Step-by-step solution

Use properties of limits

We will start by using the properties of limits to transform the given limit into a more manageable form. Recall that if \(\lim_{n\rightarrow\infty} a_n = a\) and \(\lim_{n\rightarrow\infty} b_n = b\), then $$\lim_{n\rightarrow\infty} (\frac{a_n}{b_n}) = \frac{\lim_{n\rightarrow\infty} a_n}{\lim_{n\rightarrow\infty} b_n} = \frac{a}{b}$$ In our case, \(a_n = s_n\) and \(b_n = t_n\). So, we are trying to prove that $$\lim_{n\rightarrow\infty} \frac{s_n}{t_n} = \frac{\lim_{n\rightarrow\infty} s_n}{\lim_{n\rightarrow\infty} t_n} = \frac{s}{t}$$ given that \(s / t\) is defined in the extended reals and \(t \neq 0\).

Restate the given limit

We want to prove that $$\lim_{n\rightarrow\infty} \frac{s_n}{t_n} = \frac{s}{t}$$ where \(s = \lim_{n\rightarrow\infty} s_n\) and \(t = \lim_{n\rightarrow\infty} t_n\). The definition of a limit states that for any \(\epsilon > 0\), there exists \(N\) such that for all \(n > N\), we have $$\left|\frac{s_n}{t_n} - \frac{s}{t}\right| < \epsilon$$

Prove the limit using the definition of limit

To prove the limit, we need to show that for any given \(\epsilon > 0\), there is an \(N\) large enough such that $$\left|\frac{s_n}{t_n} - \frac{s}{t}\right| < \epsilon$$ for all \(n > N\). We can rewrite the expression as $$\left|\frac{s_n}{t_n} \cdot \frac{t - t_n}{t - t_n} - \frac{s - s_n}{t - t_n} \cdot \frac{t - t_n }{ t} \right| = \left| \frac{(s_n t - s_n t_n ) - (s t_n - s_n t_n )}{t_n (t - t_n )}\right| = \left| \frac{s (t - t_n) - s_n (t - t_n)}{t_n (t - t_n )}\right|$$ Now, choose a large enough \(N\) such that for all \(n > N\), $$|s_n - s| < \frac{\epsilon |t|}{2} \quad\text{and}\quad |t_n - t| < \frac{\epsilon |s|}{2}$$ Using the triangle inequality, we can bound the expression: $$\left|\frac{s_n}{t_n}-\frac{s}{t}\right| = \left|\frac{s (t - t_n) - s_n (t - t_n)}{t_n (t - t_n)}\right| \le \left|\frac{(s - s_n)(t - t_n)}{t_n (t - t_n)}\right| \le \frac{\epsilon^2}{4} < \epsilon$$ Thus, we have proven that \(\lim_{n\rightarrow\infty} \frac{s_n}{t_n} = \frac{s}{t}\).

Provide a counterexample

Consider the sequences \(s_n = n\) and \(t_n = 2n\). We have $$\lim_{n\rightarrow\infty} s_n = \infty \quad\text{and}\quad \lim_{n\rightarrow\infty} t_n = \infty$$ Thus, \(\frac{s}{t} = \frac{\infty}{\infty}\) is undefined in the extended reals. However, the sequence \(\frac{s_n}{t_n}\) converges: $$\lim_{n\rightarrow\infty} \frac{s_n}{t_n} = \lim_{n\rightarrow\infty} \frac{n}{2n} = \lim_{n\rightarrow\infty} \frac{1}{2} = \frac{1}{2}$$ In this case, \(s/t\) is not defined in the extended reals, but the limit of the sequence converges, which means (A) does not hold.

Key concepts

Convergence of Sequences

When we talk about the convergence of sequences, we are referring to the idea that as we progress through the terms of a sequence, they get closer and closer to a specific value, also known as the limit. A sequence \(s_n\) is said to converge to a limit \(s\) if, for any given \(\epsilon > 0\), there exists a natural number \(N\) such that for all \(n > N\), the absolute distance between \(s_n\) and \(s\) is less than \(\epsilon\).
In simpler terms, no matter how small the distance we specify (that is, \(\epsilon\)), we can always find a point in the sequence after which all the terms are within that distance from \(s\). This concept is fundamental in understanding how to work with limits and analyzing the behavior of sequences as they approach infinity.

Properties of Limits

The properties of limits are a set of rules that describe how limits behave under various mathematical operations. These properties are critical in solving limit problems without having to calculate the limits from first principles each time.
Some of the essential properties include the fact that the limit of a sum is the sum of the limits, the limit of a product is the product of the limits, and the limit of a quotient is the quotient of the limits, provided that the limit in the denominator is not zero. However, these rules only apply when the limits involved are finite and when certain conditions, such as the non-zero denominator, are met. These properties are what make it possible to deconstruct more complex expressions into simpler parts whose limits can be more easily determined.

Extended Reals

The concept of extended reals is an expansion of the real number system which includes the symbols \(+\infty\) and \( -\infty\) to represent positive and negative infinity, respectively. This system allows us to discuss the behavior of sequences and functions as they grow without bounds.
In the extended real number system, we can now speak of limits that approach infinity or negative infinity, which is not possible with only the real numbers. But despite this expansion, there are still undefined operations, such as \(\frac{\infty}{\infty}\), which the extended real system does not resolve. Our textbook problem required that \(s/t\) be defined in the extended reals, exempting cases where both \(s\) and \(t\) approach infinity, since the determination of their ratio would still be indeterminate.

Counterexample in Limits

A counterexample in limits is a specific case that disproves a general statement about limits. It is a powerful tool in mathematical arguments because it demonstrates that a limit rule or property does not always apply.
For instance, we might have a situation where a property holds true for all known examples, but finding just one example where it fails is enough to show that the property is not universally valid. In our textbook problem, we provided a counterexample where the sequence \(s_n\) and \(t_n\) both approach infinity while their ratio converges to a finite limit. This particular scenario demonstrates that the property stating the limit of a quotient is the quotient of the limits, under specific circumstances, does not hold when the limits are extended to include infinity, which is crucial in understanding the delimitations of these mathematical principles.