y(0) y(x)=x+1/x what happens to the function as x increases? what happens to the function when x decreases

The function given is:

\[
y(x) = x + \frac{1}{x}
\]

To analyze what happens to this function as \( x \) increases or decreases, we can consider two cases:

### 1. **As \( x \) increases (\( x \to \infty \)):**

- The term \( x \) will increase without bound.
- The term \( \frac{1}{x} \) will decrease towards 0 because the reciprocal of a large number approaches zero.

Therefore, as \( x \to \infty \), the function \( y(x) \approx x + 0 = x \). This implies that \( y(x) \) will increase without bound as \( x \) increases. In other words, \( y(x) \) will go to infinity as \( x \) goes to infinity.

### 2. **As \( x \) decreases (\( x \to 0^+ \) from the positive side):**

- The term \( x \) approaches 0.
- The term \( \frac{1}{x} \) becomes very large because the reciprocal of a small positive number is a large number.

Therefore, as \( x \to 0^+ \), the function \( y(x) \approx 0 + \frac{1}{x} \). This implies that \( y(x) \) will go to infinity as \( x \) approaches 0 from the positive side.

### 3. **As \( x \) decreases (\( x \to 0^- \) from the negative side):**

- The term \( x \) approaches 0 (but from the negative side).
- The term \( \frac{1}{x} \) becomes very large but negative because the reciprocal of a small negative number is a large negative number.

Therefore, as \( x \to 0^- \), the function \( y(x) \approx 0 + \frac{1}{x} \). This implies that \( y(x) \) will go to negative infinity as \( x \) approaches 0 from the negative side.

### 4. **As \( x \) decreases (\( x \to -\infty \)):**

- The term \( x \) will decrease without bound, becoming a large negative number.
- The term \( \frac{1}{x} \) will also approach 0 but remain negative because the reciprocal of a large negative number is a small negative number.

Therefore, as \( x \to -\infty \), the function \( y(x) \approx x + 0 = x \). This implies that \( y(x) \) will decrease without bound as \( x \) decreases, approaching negative infinity.

### Summary:
- As \( x \to \infty \), \( y(x) \to \infty \).
- As \( x \to 0^+ \), \( y(x) \to \infty \).
- As \( x \to 0^- \), \( y(x) \to -\infty \).
- As \( x \to -\infty \), \( y(x) \to -\infty \).

Would you like more details or have any questions?

### Related Questions:
1. What is the derivative of the function \( y(x) = x + \frac{1}{x} \)?
2. How do we find the critical points of the function \( y(x) = x + \frac{1}{x} \)?
3. What is the behavior of the function near its critical points?
4. How does the function \( y(x) = x + \frac{1}{x} \) behave in the interval \( (-1, 1) \)?
5. What are the asymptotes of the function \( y(x) = x + \frac{1}{x} \)?

### Tip:
To analyze the behavior of a function as \( x \) increases or decreases, look at the limits of each term separately and see how they contribute to the overall behavior of the function.

Explore the behavior of the function y(x) = x + 1/x as x increases to infinity and decreases to zero and negative infinity. Understand how the function approaches positive and negative infinity in different intervals. Suitable for advanced high school students and beyond.

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