as x increases, what happens numerically to the term 1/x? y(0) y(x)=x+1/x

Let's break down the problem.

### Analysis of the Term \( \frac{1}{x} \):
As \( x \) increases, the term \( \frac{1}{x} \) decreases. This is because \( \frac{1}{x} \) is a hyperbolic function, which means as \( x \) gets larger, the value of \( \frac{1}{x} \) approaches zero. Specifically:
- When \( x \) is small (but positive), \( \frac{1}{x} \) is large.
- When \( x \) increases, \( \frac{1}{x} \) gets smaller and smaller.
- As \( x \) approaches infinity, \( \frac{1}{x} \) approaches 0.

### Function \( y(x) = x + \frac{1}{x} \):
The function you provided is \( y(x) = x + \frac{1}{x} \).

#### Behavior of \( y(x) \) as \( x \) Increases:
- **For very small \( x \)** (close to 0 but positive): \( \frac{1}{x} \) dominates, so \( y(x) \) is very large.
- **For intermediate \( x \)**: \( x \) and \( \frac{1}{x} \) contribute, with \( x \) gradually becoming more dominant as \( x \) increases.
- **For large \( x \)**: \( x \) dominates, and \( \frac{1}{x} \) becomes negligible, so \( y(x) \approx x \).

### Summary:
- As \( x \) increases, the term \( \frac{1}{x} \) decreases and approaches 0.
- Consequently, \( y(x) = x + \frac{1}{x} \) increases approximately linearly, approaching \( y(x) \approx x \) for large \( x \).

Would you like further details or have any questions?

Here are some related questions to deepen your understanding:

1. What is the behavior of \( y(x) = x + \frac{1}{x} \) as \( x \) approaches 0 from the right?
2. What is the minimum value of \( y(x) = x + \frac{1}{x} \) and at what \( x \) does it occur?
3. How does the derivative of \( y(x) = x + \frac{1}{x} \) change with respect to \( x \)?
4. What happens to the function \( y(x) = x + \frac{1}{x} \) if \( x \) is allowed to be negative?
5. How would the graph of \( y(x) = x + \frac{1}{x} \) look for various ranges of \( x \)?

**Tip:** Understanding the behavior of functions as \( x \) approaches certain critical points (like 0 or infinity) is key to analyzing their overall behavior.

Explore how the term 1/x behaves as x increases in the function y(x) = x + 1/x. Understand the relationship between x and 1/x, and how it impacts the function's behavior. Suitable for high school students studying functions and their properties.

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