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

Question: Prove that \(\lim_{x \rightarrow \infty} \frac{10}{x} = 0\) using the definition of a limit. Answer: 1. Set up the inequality: \(\left|\frac{10}{x}\right| < \varepsilon\) 2. Find a suitable value for N: \(N = \frac{10}{\varepsilon}\) 3. Show the inequality holds for chosen N: \(\left|\frac{10}{x}\right| < \varepsilon\) when \(x > \frac{10}{\varepsilon}\) Thus, we have proven that \(\lim_{x \rightarrow \infty} \frac{10}{x} = 0\).

Step-by-step solution

Set up the inequality

We want to prove that for any \(\varepsilon > 0\), there exists an N such that when \(x > N\), the following inequality holds: $$\left|\frac{10}{x}\right| < \varepsilon$$

Find a suitable value for N

In order to prove the limit statement, we need to find a value of N that satisfies the given inequality. We are given that \(x > N\), so we can try to manipulate the inequality to obtain an inequality involving N. Start by dividing both sides of the inequality by 10: $$\frac{1}{x} < \frac{\varepsilon}{10}$$ Now, take the reciprocal of both sides. Since \(x > N > 0\), we do not change the direction of the inequality: $$x > \frac{10}{\varepsilon}$$ We can choose the value of N as \(N = \frac{10}{\varepsilon}\).

Show the inequality holds for chosen N

Now that we have chosen a value for N, we need to show that the inequality holds when \(x > N\). So we have: $$x > \frac{10}{\varepsilon}$$ Taking the reciprocal of both sides (since \(x > 0\) and \(\varepsilon > 0\)): $$\frac{1}{x} < \frac{\varepsilon}{10}$$ Multiplying both sides by 10: $$\left|\frac{10}{x}\right| < \varepsilon$$ This proves that \(\lim_{x \rightarrow \infty} \frac{10}{x} = 0\).

Key concepts

Epsilon-Delta Definition

The epsilon-delta definition is a formal way to prove that a function approaches a certain limit as the variable approaches infinity. In terms of limits at infinity, we use this precise definition to mathematically show what happens to the function as it grows larger and larger.

The basic idea is: we want to make sure that the function value gets arbitrarily close to the limit (called \(L\)) as the variable \(x\) gets larger. To do this, we specify a small number \(\varepsilon > 0\) (epsilon represents how close we want to be to the limit).

• For every \(\varepsilon\), however small, there must exist a corresponding \(N > 0\) (delta) such that whenever \(x > N\), the expression \(|f(x) - L| < \varepsilon\) holds true.

This means the difference between the function value and \(L\) is smaller than \(\varepsilon\), ensuring the function stays close to \(L\) once \(x\) exceeds \(N\). This method demands a logical construction of \(N\) in terms of \(\varepsilon\) to verify the limit exists and equals the given value.

Asymptotic Behavior

Asymptotic behavior refers to how functions behave as the input values become very large (approaching infinity) or very small. Observing asymptotic behavior is essential for understanding limits at infinity because it tells us the end behavior of functions.

In particular cases, such as the limit \(\lim_{x \to \infty} \frac{10}{x} = 0\), the function approaches a horizontal line, known as an asymptote. Here, \(y = 0\) is the horizontal asymptote because as \(x\) grows, \(\frac{10}{x}\) yields smaller values, getting closer to zero.

• The function's distance from the asymptote decreases as \(x\) increases.
• Knowing the asymptotic behavior helps predict the function's trend.

Being mindful of asymptotes highlights the function's limits and continuity with respect to growing or diminishing values. Evaluating this behavior allows us to contextualize the limit and ensure it aligns with expected outcomes when modeling scenarios in vast scales, such as physics or economics.

Reciprocal Functions

Reciprocal functions are a fascinating category of functions where the function is defined as the reciprocal of a given function. In our context, we examine how reciprocals behave as their inputs become increasingly large.

The basic form of a reciprocal function is \(f(x) = \frac{k}{x}\), where \(k\) is a constant. As \(x\) increases towards infinity, \(\frac{k}{x}\) diminishes since a larger denominator results in a smaller fraction.

• When \(x\) becomes infinitely large, the reciprocal function approaches zero.
• This is why limits involving expressions like \(\frac{10}{x}\) lead towards zero as \(x\) heads to infinity.

This innate tendency of reciprocal functions to decline towards zero serves as the foundation for proving limits at infinity using the epsilon-delta definition. Understanding this core behavior assists in visualizing why and how the function's outputs shrink, leading to zero as the inputs extend indefinitely.