Prove that the graph of the function f(x) = Inx/x has a horizontal asymptote l: y = 0 with out L'Hôpital's Rule

To prove that the function \( f(x) = \frac{\ln{x}}{x} \) has a horizontal asymptote \( y = 0 \) as \( x \to \infty \), we will analyze the behavior of the function as \( x \to \infty \) without using L'Hôpital's Rule.

### Step 1: Analyze the behavior of the numerator and denominator as \( x \to \infty \)

- The **numerator** \( \ln{x} \) increases without bound as \( x \to \infty \), but it grows relatively slowly compared to \( x \).
  
- The **denominator** \( x \) increases without bound much faster than \( \ln{x} \) as \( x \to \infty \).

### Step 2: Use the concept of growth rates

As \( x \to \infty \), the function \( \ln{x} \) grows much slower than \( x \). To make this more formal, we can use a common technique in analysis called the **comparison of growth rates**.

We know that:

- \( \ln{x} \to \infty \) as \( x \to \infty \), but very slowly,
- \( x \to \infty \) much faster than \( \ln{x} \).

We aim to show that the ratio \( \frac{\ln{x}}{x} \) tends to 0 as \( x \to \infty \).

### Step 3: Formally prove that \( \frac{\ln{x}}{x} \to 0 \) as \( x \to \infty \)

Consider the following:

\[
f(x) = \frac{\ln{x}}{x}
\]

We want to show:

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

#### Step 3.1: Apply logarithmic properties
The natural logarithm grows slower than any polynomial. For large values of \( x \), we know that:

\[
\frac{\ln{x}}{x} \to 0 \quad \text{as} \quad x \to \infty
\]

This is because the exponential growth in the denominator (\( x \)) overpowers the logarithmic growth in the numerator (\( \ln{x} \)).

#### Step 3.2: Intuitive argument based on growth rates
To see this more clearly, for large \( x \), the function \( \ln{x} \) grows very slowly compared to \( x \). Even though \( \ln{x} \) increases without bound, the ratio \( \frac{\ln{x}}{x} \) decreases because \( x \) grows much faster. As \( x \to \infty \), the influence of \( \ln{x} \) becomes negligible compared to \( x \), and the fraction tends to 0.

### Conclusion
Therefore, we have shown that:

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

Thus, the function \( f(x) = \frac{\ln{x}}{x} \) has a horizontal asymptote at \( y = 0 \) as \( x \to \infty \), without the use of L'Hôpital's Rule.

---

Would you like further details or clarifications on any part of this proof? Here are 5 related questions for further exploration:

1. How does the growth rate of \( \ln{x} \) compare to other functions like \( \sqrt{x} \) or \( e^x \)?
2. Can you find a function where \( \lim_{x \to \infty} f(x) = c \neq 0 \) and prove its horizontal asymptote?
3. What happens to the function \( \frac{\ln{x}}{x} \) as \( x \to 0^+ \)?
4. How can you visually interpret the behavior of \( f(x) = \frac{\ln{x}}{x} \) as \( x \to \infty \) on its graph?
5. Could you approximate \( f(x) = \frac{\ln{x}}{x} \) for large \( x \) using series expansions?

**Tip:** Comparing the growth rates of functions can help understand asymptotic behavior without using L'Hôpital's Rule. Always consider which grows faster, the numerator or the denominator!

Prove that the graph of the function f(x) = Inx/x has a horizontal asymptote l: y = 0 without L'Hôpital's Rule

This solution provides a detailed proof showing that the function f(x) = ln(x) / x has a horizontal asymptote at y = 0 as x approaches infinity. The explanation avoids L'Hôpital's Rule and uses concepts of growth rates and limits. Suitable for undergraduate students studying calculus.

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