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The problem you're referring to asks to show that the limit of the function \( f(t) \) as \( t \) approaches 2 does not exist. Let's break this down.

### Given:
The function \( f(t) \) is defined as:
\[
f(t) =
\begin{cases}
30,000, & 0 < t \leq 1 \\
32,400, & 1 < t \leq 2 \\
34,992, & 2 < t \leq 3
\end{cases}
\]
Where \( t \) represents time in years, and \( f(t) \) represents salary in thousands of dollars.

### We are asked to prove:
The limit \( \lim_{t \to 2} f(t) \) does not exist.

### Steps:
To determine if the limit exists, we must check the left-hand limit (\( \lim_{t \to 2^-} f(t) \)) and the right-hand limit (\( \lim_{t \to 2^+} f(t) \)):

1. **Left-hand limit \( \lim_{t \to 2^-} f(t) \)**:
   - For \( 1 < t \leq 2 \), the value of \( f(t) \) is 32,400 (in thousands).
   - As \( t \) approaches 2 from the left, \( f(t) = 32,400 \).
   - Thus, \( \lim_{t \to 2^-} f(t) = 32,400 \).

2. **Right-hand limit \( \lim_{t \to 2^+} f(t) \)**:
   - For \( 2 < t \leq 3 \), the value of \( f(t) \) is 34,992 (in thousands).
   - As \( t \) approaches 2 from the right, \( f(t) = 34,992 \).
   - Thus, \( \lim_{t \to 2^+} f(t) = 34,992 \).

### Conclusion:
Since \( \lim_{t \to 2^-} f(t) = 32,400 \) and \( \lim_{t \to 2^+} f(t) = 34,992 \), the left-hand limit and right-hand limit are not equal. Therefore, the limit \( \lim_{t \to 2} f(t) \) **does not exist**.

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Do you want further details on this, or have any questions?

Here are 5 related questions:
1. What is the general definition of a limit?
2. How does the concept of continuity relate to limits?
3. Can the function \( f(t) \) be made continuous by adjusting its values?
4. What happens if the left and right-hand limits are equal?
5. What is the significance of piecewise functions in real-world problems like this?

**Tip:** For a limit to exist at a point, both the left-hand and right-hand limits must be equal to the same value.

Show that the limit of f(t) as t approaches 2 does not exist. The function f(t) is piecewise defined as: 30,000 for 0 < t ≤ 1, 32,400 for 1 < t ≤ 2, and 34,992 for 2 < t ≤ 3.

This problem demonstrates why the limit of a piecewise function does not exist at a specific point when the left-hand and right-hand limits differ. We analyze the function f(t) with distinct salary values over different time intervals and prove that the limit as t approaches 2 does not exist.

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