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:

\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.