Let's break down the problem from the image you provided.

The task asks you to:

1. Graph the function $f(x) = |x|$.
2. Find the following limits:
   • $\lim_{x \to 0} |x|$
   • $\lim_{x \to 1} |x|$

1. Graphing $f(x) = |x|$

The function $f(x) = |x|$ represents the absolute value of $x$. The graph is a V-shape:

• For $x \geq 0$, $|x| = x$.
• For $x < 0$, $|x| = -x$.

This means the graph has a sharp corner at $x = 0$, and it is symmetric about the y-axis.

2. Finding the Limits

(a) $\lim_{x \to 0} |x|$

To find this limit, we evaluate the behavior of $|x|$ as $x$ approaches 0 from both directions:

• As $x \to 0^+$ (from the right), $|x| = x$, and $x \to 0$.
• As $x \to 0^-$ (from the left), $|x| = -x$, and $-x \to 0$.

Thus, the two-sided limit exists, and $\lim_{x \to 0} |x| = 0$.

(b) $\lim_{x \to 1} |x|$

Here we consider the behavior of $|x|$ as $x$ approaches 1:

• As $x \to 1^+$, $|x| = x = 1$.
• As $x \to 1^-$, $|x| = -x = 1$ (since the absolute value is still positive).

Therefore, $\lim_{x \to 1} |x| = 1$.

Would you like further clarification or additional details on the graph or limits?

Here are 5 related questions to help deepen your understanding:

1. How would the graph of $f(x) = |x|$ change if it was $f(x) = |x-2|$?
2. What is the difference between a one-sided limit and a two-sided limit?
3. How does the sharp corner at $x = 0$ affect the differentiability of $f(x) = |x|$?
4. Can you think of another function with a sharp corner like $f(x) = |x|$?
5. How would you compute $\lim_{x \to -1} |x|$?

Tip: When dealing with limits, always consider the behavior from both the left-hand and right-hand sides to ensure the two-sided limit exists.