State whether the following statements are True of False.
`lim_(xto3)|x|` exists and equal to 3.

Answer

To determine whether the statement `lim_(x→3)|x|` exists and is equal to 3, we will analyze the limit step by step.

### Step-by-Step Solution:

1. **Understanding the Limit**: We need to find the limit of |x| as x approaches 3. The expression can be written as:
\[ \lim_{x \to 3} |x| \]

2. **Finding Left-Hand Limit**: We will first find the left-hand limit as x approaches 3 from the left (i.e., x approaches 3 with values less than 3). We can express this as:
\[ \lim_{x \to 3^-} |x| = \lim_{h \to 0} |3 - h| \]
Here, h is a small positive number, so:
\[ |3 - h| = 3 - h \]
As h approaches 0, this limit becomes:
\[ \lim_{h \to 0} (3 - h) = 3 \]

3. **Finding Right-Hand Limit**: Next, we find the right-hand limit as x approaches 3 from the right (i.e., x approaches 3 with values greater than 3). This can be expressed as:
\[ \lim_{x \to 3^+} |x| = \lim_{h \to 0} |3 + h| \]
Here, h is again a small positive number, so:
\[ |3 + h| = 3 + h \]
As h approaches 0, this limit becomes:
\[ \lim_{h \to 0} (3 + h) = 3 \]

4. **Comparing Left-Hand and Right-Hand Limits**: Now we compare the left-hand limit and the right-hand limit:
- Left-hand limit: 3
- Right-hand limit: 3

Since both limits are equal, we conclude that:
\[ \lim_{x \to 3} |x| = 3 \]

5. **Conclusion**: Since the limit exists and is equal to 3, we can state that the original statement is **True**.

### Final Answer:
The statement `lim_(x→3)|x|` exists and is equal to 3 is **True**.