https://file.aiquickdraw.com/mathbot/file/172703965192733x2ubp6.jpg

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, \( \lim_{x \to 1} |x| = 1 \).

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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.

Graph the function f(x) = |x| and find the limits lim(x -> 0) |x| and lim(x -> 1) |x|.

Practice graphing the absolute value function f(x) = |x| and finding limits such as lim(x -> 0) |x| and lim(x -> 1) |x|. Learn how to analyze both continuous and discontinuous functions graphically and numerically, suitable for high school students in grades 10-12.

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