To determine if $f(x) = |x + 3|$ is differentiable at $x = -3$, we use the alternate definition of a derivative:

$f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$

In this case, $c = -3$, so the expression becomes:

$f'(-3) = \lim_{x \to -3} \frac{f(x) - f(-3)}{x + 3}$

Step 1: Evaluate $f(-3)$

We know that $f(x) = |x + 3|$. At $x = -3$:

$f(-3) = |-3 + 3| = |0| = 0$

Thus, $f(-3) = 0$.

Step 2: Set up the difference quotient

We now need to analyze the difference quotient on both sides of $x = -3$:

$f'( -3 ) = \lim_{x \to -3} \frac{|x + 3| - 0}{x + 3} = \lim_{x \to -3} \frac{|x + 3|}{x + 3}$

Step 3: Left-hand limit ($x \to -3^-$)

When $x \to -3^-$ (approaching from the left), $x + 3$ is negative. Therefore, $|x + 3| = -(x + 3)$.

Thus, the difference quotient becomes:

$\lim_{x \to -3^-} \frac{-(x + 3)}{x + 3} = \lim_{x \to -3^-} -1 = -1$

Step 4: Right-hand limit ($x \to -3^+$)

When $x \to -3^+$ (approaching from the right), $x + 3$ is positive. Therefore, $|x + 3| = (x + 3)$.

Thus, the difference quotient becomes:

$\lim_{x \to -3^+} \frac{x + 3}{x + 3} = \lim_{x \to -3^+} 1 = 1$

Step 5: Conclusion

Since the left-hand limit ($-1$) and the right-hand limit ($1$) are not equal, the derivative does not exist at $x = -3$. Therefore, $f(x) = |x + 3|$ is not differentiable at $x = -3$.

Key Reason:

The absolute value function $|x + 3|$ introduces a sharp corner at $x = -3$, causing a discontinuity in the slope from either side, preventing the derivative from existing at that point.

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Would you like further clarification, or do you have any questions?

Here are 5 related questions to consider:

1. How does the absolute value function affect differentiability?
2. What are the conditions for differentiability at a point?
3. Can a function be continuous but not differentiable at a point? Why?
4. What is the geometric interpretation of non-differentiability at sharp corners?
5. How can the piecewise definition of absolute value functions help in calculating derivatives?

Tip: Always check both left-hand and right-hand limits when verifying the existence of a derivative at a point, especially for piecewise or absolute value functions.