To show that the function $f(x) = |x|$ is not differentiable at $x = 0$, we need to check if the limit of the difference quotient exists at this point. The difference quotient is given by:

$\lim_{h \to 0} \frac{f(0+h) - f(0)}{h}$

Step 1: Define the function $f(x)$

The function is:

$f(x) = |x| \quad \text{for all} \, x \in \mathbb{R}$

At $x = 0$, we have:

$f(0) = |0| = 0$

Thus, we need to examine the limit:

$\lim_{h \to 0} \frac{|h| - 0}{h} = \lim_{h \to 0} \frac{|h|}{h}$

Step 2: Compute the limit from both sides of $h$

We need to evaluate the limit of $\frac{|h|}{h}$ as $h \to 0$ from both the positive and negative sides.

• When $h > 0$: In this case, $|h| = h$, so:

  $\frac{|h|}{h} = \frac{h}{h} = 1$

• When $h < 0$: In this case, $|h| = -h$, so:

  $\frac{|h|}{h} = \frac{-h}{h} = -1$

Step 3: Conclusion

The limit from the right (when $h > 0$) is 1, and the limit from the left (when $h < 0$) is -1. Since these two one-sided limits are not equal, the limit does not exist at $h = 0$.

Thus, the function $f(x) = |x|$ is not differentiable at $x = 0$.

Recap:

The function $f(x) = |x|$ is not differentiable at $x = 0$ because the left-hand and right-hand limits of the difference quotient are not equal.

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Would you like more details or further clarification? Here are some related questions to explore:

1. What does it mean for a function to be differentiable at a point?
2. How do we determine differentiability at points other than $x = 0$ for piecewise functions like $f(x) = |x|$?
3. What is the geometric interpretation of a non-differentiable point in the graph of a function?
4. How does the concept of continuity relate to differentiability?
5. Can a function be continuous but not differentiable? Provide examples.

Tip: For piecewise functions, always check the limits of the difference quotient from both sides to determine differentiability.