Problem 64 - Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Based on the definition of a limit at infinity, we were asked to prove that the limit of the function \(f(x) = \frac{10}{x}\) as x approaches infinity is 0. We showed that for any \(\varepsilon > 0\), there exists an \(N > 0\) such that \(|f(x) - 0| < \varepsilon\) whenever \(x > N\). We found the suitable value for \(N\) as \(N = \frac{10}{\varepsilon}\), and by verifying this condition, we concluded that the limit of the function as x approaches infinity is indeed 0: $$\lim_{x \rightarrow \infty} \frac{10}{x} = 0$$

Step 1: Rewrite the Absolute Value of the Difference

As we want to prove the limit is 0 using the definition, we will rewrite the expression \(|f(x) - 0|\) for the given function \(f(x) = \frac{10}{x}\): $$|f(x) - 0| = \left| \frac{10}{x} - 0 \right| = \left| \frac{10}{x} \right|$$

Step 2: Find a Suitable N

We need to find a suitable value of \(N > 0\) such that the inequality \(\left| \frac{10}{x} \right| < \varepsilon\) holds true for all \(x > N\). We can relate \(\varepsilon\) and \(N\) as follows: $$\left| \frac{10}{x} \right| < \varepsilon \Longrightarrow \frac{10}{x} < \varepsilon \Longrightarrow x > \frac{10}{\varepsilon}$$ By this inequality, we can see that if we choose \(N = \frac{10}{\varepsilon}\), then for all \(x > N\), the inequality \(\left| \frac{10}{x} \right| < \varepsilon\) will hold.

Step 3: Verify the Condition

Now that we have found a suitable value for \(N\), let's verify that this value of \(N\) satisfies the condition of the definition of the limit at infinity: Let \(x > N = \frac{10}{\varepsilon}\). Then: $$\left| \frac{10}{x} \right| = \frac{10}{x} < \frac{10}{\frac{10}{\varepsilon}} = \varepsilon$$ So, when \(x > \frac{10}{\varepsilon}\), the inequality \(\left| \frac{10}{x} \right| < \varepsilon\) is satisfied.

Step 4: Conclusion

In conclusion, we used the definition of a limit at infinity to show that for any \(\varepsilon > 0\), we were able to find a value for \(N = \frac{10}{\varepsilon}\) such that \(|f(x) - 0| < \varepsilon\) whenever \(x > N\). Therefore, by definition, we have proven that: $$\lim_{x \rightarrow \infty} \frac{10}{x} = 0$$

Key Concepts

Epsilon-Delta Definition

The epsilon-delta definition is a formal mathematical way to describe the behavior of functions as they get close to a particular value or grow infinitely large. In the context of a limit at infinity, such as with \[ \lim_{x \to \infty} f(x) = L \], it means that as \( x \) becomes larger and larger, the function \( f(x) \) gets increasingly close to the value \( L \).
The statement "for any \( \varepsilon > 0 \), there exists \( N > 0 \) such that \( |f(x) - L| < \varepsilon \) whenever \( x > N \)" ensures that we can make \( f(x) \) as close to \( L \) as desired by choosing a sufficiently large \( N \).

• If \( \varepsilon \) is small, it indicates a tighter vicinity around \( L \).
• Choosing \( N \) based on \( \varepsilon \) ensures \( f(x) \) falls within this small range for all values of \( x\) larger than \( N \).

This concept offers mathematicians a precise tool to verify limits, laying a foundation for rigorous mathematical proofs.

Continuous Functions

Continuous functions are a fundamental concept in calculus and advanced mathematics, frequently associated with limits. A function \( f(x) \) is continuous at a point \( a \) if the following three conditions are met:

• The function \( f(x) \) is defined at \( a \).
• The limit \( \lim_{x \to a} f(x) \) exists.
• The limit at \( x = a \) matches the function value — \( \lim_{x \to a} f(x) = f(a) \).

For infinite limits, continuity ensures that no gaps, jumps, or infinite oscillations occur as \( x \) increases or decreases without bound. For instance, in the expression \( \lim_{x \to \infty} \frac{10}{x} = 0 \), the function \( f(x) = \frac{10}{x} \) is continuous, indicating that as \( x \to \infty \), the value of \( f(x) \) smoothly approaches zero without any disruption. Continuous functions are critical in calculus because they simplify the calculation of limits, derivatives, and integrals.

Proof of Limits

Proving limits, especially at infinity, involves demonstrating that a function behaves in a predictable way as \( x \) approaches infinity. To prove a limit such as \( \lim_{x \to \infty} \frac{10}{x} = 0 \), mathematicians utilize the epsilon-delta definition to establish this precise behavior.
Here's how it was shown:

• We expressed the difference \( |f(x) - 0| \) as \( \left| \frac{10}{x} \right| \).
• We then had to make \( \left| \frac{10}{x} \right| < \varepsilon \) for any given \( \varepsilon > 0 \).
• By choosing \( N = \frac{10}{\varepsilon} \), we made sure that the condition \( x > N \) guarantees the difference is less than \( \varepsilon \).

This proof technique reassures us that regardless of how small \( \varepsilon \) is, we can find a sufficiently large \( N \) for which \( x > N \) holds true, ensuring the function complies with the limit definition.