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The function \( f(x) \) is defined piecewise, and you are asking for the limit of \( f(x) \) as \( x \to -5^- \), which means we approach \( -5 \) from the left-hand side.

According to the piecewise function:

\[
f(x) = \begin{cases} 
17 & \text{if } x \leq -5 \\
-x + 12 & \text{if } -5 \leq x < 7 \\
7 & \text{if } x = 7 \\
12e^{x-7} & \text{if } x > 7
\end{cases}
\]

For \( x \to -5^- \), we are approaching from the left, so we use the case where \( x \leq -5 \). In this case, \( f(x) = 17 \) for all \( x \leq -5 \).

Thus,

\[
\lim_{x \to -5^-} f(x) = 17.
\]

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. What is the limit of \( f(x) \) as \( x \to -5^+ \)?
2. Is \( f(x) \) continuous at \( x = -5 \)?
3. What is the limit of \( f(x) \) as \( x \to 7^- \)?
4. What is \( f'(x) \) for \( -5 \leq x < 7 \)?
5. What is the value of \( \lim_{x \to +\infty} f(x) \)?

**Tip:** For piecewise functions, always pay attention to which interval your \( x \)-value belongs to when taking limits or evaluating the function at specific points.

Understand how to find the limit of the piecewise function f(x) as x approaches -5 from the left-hand side. This problem covers essential calculus concepts like limits, continuity, and piecewise functions, suitable for high school calculus students.

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