Question

The limit at infinity \(\lim _{x \rightarrow \infty} f(x)=L\) means that for any \(\varepsilon>0\) there exists \(N>0\) such that $$|f(x)-L|<\varepsilon \quad \text { whenever } \quad x>N.$$ Use this definition to prove the following statements. $$\lim _{x \rightarrow \infty} \frac{10}{x}=0$$

Short answer

After analyzing the given function \(f(x) = \frac{10}{x}\), we demonstrated that for any \(\varepsilon > 0\), there exists \(N = \frac{10}{\varepsilon} > 0\) such that \(|f(x) - 0| < \varepsilon\) whenever \(x > N\). Thus, we've proven that the limit \(\lim_{x \rightarrow \infty} \frac{10}{x} = 0\) holds true.

Step-by-step solution

1: Show the function in question

We're given the function $$f(x) = \frac{10}{x}$$ and we want to prove that $$\lim _{x \rightarrow \infty} \frac{10}{x}=0$$ using the definition of limits at infinity.

2: Write down the expression to be analyzed

We have to find \(N\) such that for all \(x > N\), $$|f(x) - 0| < \varepsilon$$ We can rewrite this expression as follows: $$\left|\frac{10}{x}\right| < \varepsilon$$

3: Determine a relationship between x and N

Now we want to find out how \(x\) and \(N\) are related. The expression we want to satisfy is: $$\frac{10}{x} < \varepsilon.$$ Let's solve for \(x\). $$\begin{align*} \frac{10}{x} &< \varepsilon \\ x &> \frac{10}{\varepsilon} \end{align*}$$ To satisfy the given inequality, \(x\) must be greater than \(\frac{10}{\varepsilon}\) which means we can let \(N = \frac{10}{\varepsilon}\).

4: Show that our choice of N satisfies the requirement

Given any \(\varepsilon > 0\), let \(N = \frac{10}{\varepsilon}\). Then, for any \(x > N\), we have: $$\begin{align*} x &> \frac{10}{\varepsilon} \\ \frac{10}{x} &< \varepsilon \end{align*}$$ This inequality holds true, which means that the definition of the limit at infinity is satisfied.

5: Conclusion

We have shown that for any \(\varepsilon>0\), there exists \(N = \frac{10}{\varepsilon} > 0\) such that $$\left|\frac{10}{x}\right| < \varepsilon$$ whenever \(x > N\). So, we can conclude that $$\lim _{x \rightarrow \infty} \frac{10}{x}=0$$ as required.

Key concepts

Limit Definition

Understanding limits, particularly limits at infinity, is crucial in calculus. When we talk about the limit of a function as it approaches infinity, we are examining the behavior of the function values as the input grows larger and larger. In the given example, we are interested in finding the limit of \(f(x) = \frac{10}{x}\) as \(x\) approaches infinity.
Here, the limit definition states that the limit \(\lim _{x \rightarrow \infty} f(x)=L\) means for every positive number \(\varepsilon\), there exists a number \(N\) such that whenever \(x > N\), the absolute difference \(|f(x) - L|\) is less than \(\varepsilon\).

For the problem at hand, this translates to ensuring that \(\left|\frac{10}{x}\right| < \varepsilon\) whenever \(x > N\). The goal here is to show that as \(x\) becomes very large, \(\frac{10}{x}\) gets arbitrarily close to \(0\), which is our limit \(L\). This is integral to understanding how functions behave at infinity, highlighting their tendency to settle at a particular value or approach zero.

Epsilon-Delta Proof

The epsilon-delta method is a formal approach used to prove the limit of a function, and it even extends to limits at infinity. In our scenario, we have been provided with a function \(f(x) = \frac{10}{x}\), and we aim to establish that the limit as \(x\) approaches infinity is \(0\).

To demonstrate this limit using epsilon-delta proof, first, specify an arbitrary \(\varepsilon > 0\). Our task is to find a corresponding \(N\) such that for all \(x > N\), the inequality \(|f(x) - L| < \varepsilon\) holds true.

In this context, we need \(|f(x) - 0| < \varepsilon\), which simplifies to \(\left|\frac{10}{x}\right| < \varepsilon\). Solving this inequality involves isolating \(x\), by manipulating \(\frac{10}{x} < \varepsilon\) and finding \(x > \frac{10}{\varepsilon}\).

By selecting \(N = \frac{10}{\varepsilon}\), we ensure whenever \(x > N\), \(\frac{10}{x} < \varepsilon\), thereby satisfying the limit condition for any chosen \(\varepsilon\). This method essentially "boxes in" the function's behavior as \(x\) increases beyond \(N\), confirming the function approaches the desired limit.

Convergence

Convergence is a core concept when dealing with limits, particularly as it pertains to sequences or functions approaching a certain value. In our example, the function \(f(x) = \frac{10}{x}\) exhibits convergence as \(x\) grows larger.

Convergence to a limit means that as you continue increasing \(x\) without bound, the function values \(f(x)\) move increasingly close to a specific number, here \(0\). This implies that the values of \(\frac{10}{x}\), as \(x\) becomes very large, get arbitrarily close to \(0\), indicating that the function converges to \(0\).

The principle of convergence helps visualize how a function stabilizes its behavior at infinity. Despite starting values fluctuating, by the time \(x\) tends to be significantly large, \(\frac{10}{x}\) consistently approximates \(0\).

Understanding convergence enables learners to predict function values in real-world scenarios, especially in terms of stability and expectation over large scales. It is a powerful tool in calculus and mathematical analysis to confirm the limits of various functions.