https://file.aiquickdraw.com/mathbot/file/172712031822527h4ivlf.jpg

Let's analyze the function and compute the requested limits step-by-step. The piecewise function is:

\[
f(x) = \begin{cases} 
\frac{5}{x}, & \text{if } x < 0 \\
x^2 - 3x, & \text{if } 0 \leq x < 4 \\
3x, & \text{if } x \geq 4
\end{cases}
\]

### Part (a): \(\lim_{x \to 0^-} f(x)\)
For \(x \to 0^-\), we are approaching zero from the left side (where \(x < 0\)). In this case, the relevant piece of the function is \(f(x) = \frac{5}{x}\).

\[
\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} \frac{5}{x} = -\infty
\]
Because as \(x\) approaches 0 from the negative side, \(\frac{5}{x}\) tends to \(-\infty\).

### Part (b): \(\lim_{x \to 0^+} f(x)\)
For \(x \to 0^+\), we are approaching zero from the right side (where \(x \geq 0\)). In this case, the relevant piece of the function is \(f(x) = x^2 - 3x\).

\[
\lim_{x \to 0^+} f(x) = (0)^2 - 3(0) = 0
\]

### Part (c): \(\lim_{x \to 3} f(x)\)
At \(x = 3\), we use the middle piece of the function, \(f(x) = x^2 - 3x\), because \(0 \leq x < 4\).

\[
f(3) = (3)^2 - 3(3) = 9 - 9 = 0
\]
Thus, the limit is:
\[
\lim_{x \to 3} f(x) = 0
\]

### Final Answers:
- (a) \(\lim_{x \to 0^-} f(x) = -\infty\)
- (b) \(\lim_{x \to 0^+} f(x) = 0\)
- (c) \(\lim_{x \to 3} f(x) = 0\)

Let me know if you'd like further details or explanations on any part of this.

Here are five related questions to expand on:
1. What happens if \(f(x)\) is evaluated at a specific point, such as \(x = -2\)?
2. How can we determine if a piecewise function is continuous at a point like \(x = 0\)?
3. What does it mean for a limit to not exist, and can this apply to any piecewise function?
4. Can a function have different limits from the left and right at

Consider the following function: f(x) = { 5/x, if x < 0; x^2 - 3x, if 0 ≤ x < 4; 3x, if x ≥ 4 }. Find: a. lim(x→0-) f(x); b. lim(x→0+) f(x); c. lim(x→3) f(x).

Learn how to find the limits of a piecewise function with a detailed step-by-step solution. This problem covers key calculus concepts, including the definition of limits and properties of limits. Suitable for students in Grades 11-12.

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