Use limit notation to describe the unbounded behavior of the given function as x approaches ∞ and as x approaches −∞.

u(x)=|x−8|+|x+6|

To describe the unbounded behavior of the function \( u(x) = |x - 8| + |x + 6| \) as \( x \to \infty \) and \( x \to -\infty \), let's analyze the function at both extremes.

### 1. As \( x \to \infty \):
- When \( x \to \infty \), both \( x - 8 \) and \( x + 6 \) become large positive values because \( x \) is much greater than 8 and 6. Therefore, the absolute values can be written as:
  \[
  |x - 8| \approx x - 8, \quad |x + 6| \approx x + 6.
  \]
- The function becomes approximately:
  \[
  u(x) \approx (x - 8) + (x + 6) = 2x - 2.
  \]
- So, as \( x \to \infty \):
  \[
  \lim_{x \to \infty} u(x) = \lim_{x \to \infty} (2x - 2) = \infty.
  \]

### 2. As \( x \to -\infty \):
- When \( x \to -\infty \), both \( x - 8 \) and \( x + 6 \) become large negative values. Therefore, the absolute values behave as:
  \[
  |x - 8| \approx -(x - 8) = -x + 8, \quad |x + 6| \approx -(x + 6) = -x - 6.
  \]
- The function becomes approximately:
  \[
  u(x) \approx (-x + 8) + (-x - 6) = -2x + 2.
  \]
- So, as \( x \to -\infty \):
  \[
  \lim_{x \to -\infty} u(x) = \lim_{x \to -\infty} (-2x + 2) = \infty.
  \]

### Conclusion:
- As \( x \to \infty \), \( u(x) \to \infty \).
- As \( x \to -\infty \), \( u(x) \to \infty \).

Thus, the function grows unbounded in both directions, and we can summarize it using limit notation as:
\[
\lim_{x \to \infty} u(x) = \infty, \quad \lim_{x \to -\infty} u(x) = \infty.
\]

Would you like more details or clarifications on any of these steps?

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**Related Questions:**
1. What are the general properties of absolute value functions?
2. How do absolute value functions behave as \( x \to 0 \)?
3. What is the significance of finding limits as \( x \to \pm\infty \) for functions?
4. Can you sketch the graph of \( u(x) = |x - 8| + |x + 6| \)?
5. How would you find the minimum value of \( u(x) \) and where it occurs?

**Tip**: When analyzing functions involving absolute values, break them down into different intervals based on where the expression inside the absolute value changes sign.

Use limit notation to describe the unbounded behavior of the given function as x approaches ∞ and as x approaches −∞.
u(x)=|x−8|+|x+6|

Explore the unbounded behavior of the function u(x) = |x - 8| + |x + 6| as x approaches ∞ and -∞. This solution uses limit notation and explains how absolute value functions behave at infinity. Suitable for students in Grades 11-12.

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