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.