https://file.aiquickdraw.com/mathbot/file/1727213103031vxsdyoxa.jpg

Let's analyze the piecewise function \( f(x) \) given in the problem:

\[
f(x) =
\begin{cases} 
\frac{6}{x}, & \text{if } x < 0 \\
x^2 - 3x, & \text{if } 0 \leq x < 2 \\
5x, & \text{if } x \geq 2 
\end{cases}
\]

We are tasked with finding the following limits:

---

### a. \( \lim_{x \to 0^-} f(x) \)

For \( x \to 0^- \), we use the function defined for \( x < 0 \), which is \( f(x) = \frac{6}{x} \).

\[
\lim_{x \to 0^-} \frac{6}{x} = -\infty
\]

---

### b. \( \lim_{x \to 0^+} f(x) \)

For \( x \to 0^+ \), we use the function defined for \( 0 \leq x < 2 \), which is \( f(x) = x^2 - 3x \).

\[
f(0^+) = 0^2 - 3(0) = 0
\]

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

---

### c. \( \lim_{x \to 1} f(x) \)

For \( 0 \leq x < 2 \), the function is \( f(x) = x^2 - 3x \), so we evaluate:

\[
f(1) = 1^2 - 3(1) = 1 - 3 = -2
\]

Thus:

\[
\lim_{x \to 1} f(x) = -2
\]

---

### d. Discontinuity at \( x = 2 \)?

We need to check whether the function is continuous at \( x = 2 \).

- Left-hand limit (\( x \to 2^- \)) is \( f(x) = x^2 - 3x \). Substituting \( x = 2 \):

\[
f(2^-) = 2^2 - 3(2) = 4 - 6 = -2
\]

- Right-hand limit (\( x \to 2^+ \)) is \( f(x) = 5x \). Substituting \( x = 2 \):

\[
f(2^+) = 5(2) = 10
\]

Since the left-hand and right-hand limits are not equal, there is a **jump discontinuity** at \( x = 2 \).

---

### Summary of answers:

- a. \( -\infty \)
- b. \( 0 \)
- c. \( -2 \)
- d. A jump discontinuity

---

Would you like more details on any of these calculations?

Here are five related questions to explore:
1. What are the types of discontinuities, and how do they differ?
2. How can limits be evaluated graphically for piecewise functions?
3. What is the significance of one-sided limits in piecewise functions?
4. How do you check continuity for a piecewise function at a given point?
5. How can limits involving infinity be interpreted in real-world contexts?

**Tip**: When evaluating limits of piecewise functions, always ensure you're using the correct part of the function for the given range.

Consider the following function: 

f(x) = 
(6/x, if x < 0; x^2 - 3x, if 0 <= x < 2; 5x, if x >= 2)

Find the following limits:
1. lim(x -> 0-) f(x)
2. lim(x -> 0+) f(x)
3. lim(x -> 1) f(x)
4. Identify the type of discontinuity at x = 2, if any.

Explore how to find limits for a piecewise function and identify the type of discontinuity at a given point. This problem covers key calculus concepts such as limits, continuity, and different types of discontinuities. Suitable for advanced high school or undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation