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The problem requires us to prove that:

\[
\lim_{{x \to \infty}} \frac{1}{x} = 0
\]

### Proof:

1. **Understanding the Function:**

   The function given is \( \frac{1}{x} \). As \( x \) becomes larger and larger (approaches infinity), the denominator increases while the numerator remains constant at 1.

2. **Applying Limit Properties:**

   Since \( x \) is in the denominator, the value of \( \frac{1}{x} \) becomes smaller as \( x \) increases. Formally, for any small positive number \( \epsilon > 0 \), we want to show that there exists some large number \( N \) such that whenever \( x > N \), it follows that:

   \[
   \left| \frac{1}{x} - 0 \right| < \epsilon
   \]

   This simplifies to:

   \[
   \frac{1}{x} < \epsilon
   \]

3. **Finding the Appropriate \( N \):**

   To satisfy the inequality \( \frac{1}{x} < \epsilon \), we can rewrite it as:

   \[
   x > \frac{1}{\epsilon}
   \]

   Therefore, we can choose \( N = \frac{1}{\epsilon} \). Then, for any \( x > N \), it holds that:

   \[
   \frac{1}{x} < \epsilon
   \]

4. **Conclusion:**

   Since for any \( \epsilon > 0 \), we can find such an \( N = \frac{1}{\epsilon} \) so that \( \frac{1}{x} < \epsilon \) whenever \( x > N \), we have:

   \[
   \lim_{{x \to \infty}} \frac{1}{x} = 0.
   \]

Would you like more details or have any further questions?

### Related Questions:

1. What is the limit of \( \frac{1}{x^2} \) as \( x \to \infty \)?
2. Can you explain how to find the limit of \( \frac{x}{x+1} \) as \( x \to \infty \)?
3. What happens to the limit of \( \frac{x^2}{x+1} \) as \( x \to \infty \)?
4. How do you prove that \( \lim_{{x \to -\infty}} \frac{1}{x} = 0 \)?
5. What is the limit of a function that approaches infinity more slowly, like \( \frac{\ln(x)}{x} \) as \( x \to \infty \)?

**Tip:** Remember that limits involving infinity often rely on understanding how fast functions grow or decay relative to each other.

Prove that \(\lim_{{x \to \infty}} \frac{1}{x} = 0\)

This article provides a step-by-step proof for the limit \(\lim_{{x \to \infty}} \frac{1}{x} = 0\). It covers key mathematical concepts such as limits at infinity and the behavior of reciprocal functions, suitable for high school to early college students.

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