Determine the following values. If a value does not exist, enter DNE.
7 + sin(x),
x < 0
f(x) =
-7 cos (x),0<=x<=pi
- 7 sin (x),
X> TT limx arrow 0- f(x)

We are tasked with evaluating the **limit**:

\[
\lim_{x \to 0^-} f(x)
\]

Given the piecewise definition of \( f(x) \), we need to determine which piece of the function is applicable for \( x \to 0^- \), meaning \( x \) approaches 0 from the left side (i.e., values less than 0).

### The given function is defined as follows:

\[
f(x) = 
\begin{cases} 
7 + \sin(x), & x < 0 \\
-7\cos(x), & 0 \leq x \leq \pi \\
-7\sin(x), & x > \pi
\end{cases}
\]

Since \( x \to 0^- \) implies \( x < 0 \), we focus on the first piece of the function:
\[
f(x) = 7 + \sin(x), \text{ for } x < 0.
\]

### Now, we calculate the limit:

\[
\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (7 + \sin(x))
\]

As \( x \to 0^- \), we know that \( \sin(x) \to 0 \). Thus, the limit simplifies to:

\[
\lim_{x \to 0^-} (7 + \sin(x)) = 7 + 0 = 7.
\]

### Conclusion:
\[
\lim_{x \to 0^-} f(x) = 7.
\]

Would you like a detailed explanation of limits or the function definition? Feel free to ask more questions or clarify concepts.

Here are 5 related questions:
1. What is the definition of a piecewise function?
2. How is a left-hand limit different from a right-hand limit?
3. What is the sine function's behavior near 0?
4. How do you calculate limits for trigonometric functions?
5. What are the properties of continuous functions?

**Tip:** Always check which part of a piecewise function applies when calculating limits.

Determine the following values. If a value does not exist, enter DNE. 7 + sin(x), x < 0 f(x) = -7 cos (x), 0<=x<=pi - 7 sin (x), X > TT limx arrow 0- f(x)

Learn how to evaluate the limit of a piecewise function involving trigonometric components as x approaches 0 from the left. This solution covers key concepts such as limits, sine function behavior, and piecewise function definitions. Ideal for students in Grades 11-12 or early calculus learners.

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