We are tasked with evaluating the following limit:

$\lim_{x \to 6} \frac{x + 6}{|x + 6|}$

Step-by-step solution:

The expression involves an absolute value, so we need to handle it by considering two cases based on the value of $x + 6$.

1. Case 1: $x \to 6^+$ (approaching 6 from the right)

   When $x > -6$, the term $x + 6$ is positive, meaning: $|x + 6| = x + 6$ Therefore, the expression simplifies to: $\frac{x + 6}{x + 6} = 1$ So, as $x \to 6^+$, the limit is: $\lim_{x \to 6^+} \frac{x + 6}{|x + 6|} = 1$

2. Case 2: $x \to 6^-$ (approaching 6 from the left)

   When $x < -6$, the term $x + 6$ is negative, so: $|x + 6| = -(x + 6)$ Therefore, the expression becomes: $\frac{x + 6}{-(x + 6)} = -1$ So, as $x \to 6^-$, the limit is: $\lim_{x \to 6^-} \frac{x + 6}{|x + 6|} = -1$

Conclusion:

Since the limit from the right is $1$ and the limit from the left is $-1$, the two one-sided limits do not match. Therefore, the two-sided limit does not exist.

Would you like further details or have any questions?

Here are 5 related questions for you:

1. How do absolute values affect the continuity of functions?
2. What are the conditions for a two-sided limit to exist?
3. Can we generalize this result for limits involving absolute value functions?
4. How does piecewise behavior influence limits?
5. What does it mean when a limit "does not exist" mathematically?

Tip: Always check one-sided limits separately when working with absolute value expressions or piecewise-defined functions.