For the function *f*(*x*)=|*x*+4|, evaluate the left and right limits of *f*(*x*) as *x* approaches −4.

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.

For the function f(x) = |x + 4|, evaluate the left and right limits of f(x) as x approaches −4.

Learn how to evaluate the left and right limits of the function f(x) = |x + 4| as x approaches −4. This detailed solution covers absolute value functions and limit evaluation, making it perfect for students in Grades 9-11.

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