https://file.aiquickdraw.com/mathbot/file/172698143377375hhd74a.jpg

The limit expression in the image is:

\[
\lim_{{x \to 2}} f(x) \quad \text{where} \quad f(x) = 
\begin{cases} 
x, & \text{if } x < 2 \\
x + 1, & \text{if } x \geq 2 
\end{cases}
\]

### Step-by-step solution:

We need to consider the left-hand limit (\(x \to 2^{-}\)) and the right-hand limit (\(x \to 2^{+}\)).

1. **Left-hand limit** (\(x \to 2^{-}\)):
   - For \(x < 2\), \(f(x) = x\).
   - So, as \(x \to 2^{-}\), \(f(x) \to 2\).
   
   Therefore, \[
   \lim_{{x \to 2^{-}}} f(x) = 2.
   \]

2. **Right-hand limit** (\(x \to 2^{+}\)):
   - For \(x \geq 2\), \(f(x) = x + 1\).
   - So, as \(x \to 2^{+}\), \(f(x) = 2 + 1 = 3\).
   
   Therefore, \[
   \lim_{{x \to 2^{+}}} f(x) = 3.
   \]

### Conclusion:
Since the left-hand limit is 2 and the right-hand limit is 3, and they are not equal, the limit **does not exist** at \(x = 2\).

Would you like any more details on this solution or further clarification? Here are five related questions to think about:

1. What conditions need to be met for a limit to exist at a point?
2. How does one compute one-sided limits in piecewise functions?
3. What is the significance of discontinuities in a function's limit?
4. Can the value of the function at the point \(x = 2\) affect the existence of the limit?
5. How do limits relate to the continuity of a function at a point?

**Tip:** When solving limit problems for piecewise functions, always check the behavior from both sides of the point of interest to determine if the limit exists!

Explore how to find the limit of a piecewise function as x approaches 2. This problem demonstrates key concepts in limits, one-sided limits, and the conditions for the existence of a limit. Suitable for Pre-Calculus and Calculus students.

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