To evaluate the left and right limits of $f(x) = |x + 4|$ as $x$ approaches $-4$, we need to consider the behavior of the function from both directions:

1. Right-hand limit: As $x \to -4^+$ (approaching -4 from the right), $x + 4$ will be positive (since values greater than -4 will make $x + 4 > 0$), so $|x + 4| = x + 4$. Therefore, $\lim_{x \to -4^+} |x + 4| = \lim_{x \to -4^+} (x + 4) = 0.$

2. Left-hand limit: As $x \to -4^-$ (approaching -4 from the left), $x + 4$ will be negative (since values less than -4 will make $x + 4 < 0$), so $|x + 4| = -(x + 4)$. Therefore, $\lim_{x \to -4^-} |x + 4| = \lim_{x \to -4^-} -(x + 4) = 0.$

Thus, both the left-hand and right-hand limits of $f(x)$ as $x \to -4$ are equal to 0.

Would you like further explanation or have any questions? Here are 5 related questions:

1. What happens to the limit of $f(x) = |x - 3|$ as $x \to 3$?
2. How does the absolute value affect the continuity of the function?
3. Can a function with absolute values have a jump discontinuity?
4. What is the general approach to finding limits of absolute value functions?
5. How would you evaluate the limit of $f(x) = |x - a|$ as $x$ approaches $a$?

Tip: When dealing with absolute value functions, always break them into cases based on whether the expression inside the absolute value is positive or negative.