https://file.aiquickdraw.com/mathbot/file/1727361562854ban992u5.jpg

The question presents the following piecewise function \( f(x) \):

\[
f(x) = \begin{cases} 
\frac{1}{x^2} \sin\left(\frac{1}{x}\right) & \text{if } x \neq 0 \\
0 & \text{if } x = 0 
\end{cases}
\]

It asks which of the following statements is **not true**:
- a. None of these
- b. Function is discontinuous at \( x = 0 \)
- c. Limit exists at \( x = 0 \)
- d. Limit does not exist at \( x = 0 \)

### Step-by-step Analysis:

1. **Limit as \( x \to 0 \)**:
   \[
   \lim_{x \to 0} \frac{1}{x^2} \sin\left(\frac{1}{x}\right)
   \]
   The term \( \sin\left(\frac{1}{x}\right) \) oscillates between \(-1\) and \(1\), while \( \frac{1}{x^2} \) grows without bound as \( x \to 0 \). Therefore, this limit does not exist because the oscillations do not settle to a single value.

2. **Continuity at \( x = 0 \)**:
   For a function to be continuous at \( x = 0 \), the limit as \( x \to 0 \) must equal the function value at \( x = 0 \). Since the limit does not exist, the function is **discontinuous** at \( x = 0 \).

3. **Correct Answer**:
   - **Statement b** is true: the function is discontinuous at \( x = 0 \).
   - **Statement c** is false: the limit does not exist at \( x = 0 \), which directly contradicts statement c.
   - **Statement d** is true: the limit does not exist, as shown above.

Thus, **option c** is **not true**.

Would you like more details, or do you have any further questions?

---

### Related Questions:
1. How do you prove whether a limit exists or not for a function?
2. Can you explain what oscillatory behavior means in the context of limits?
3. How can you check for continuity of a piecewise function?
4. What happens if a function has no limit but is still bounded?
5. Why do some limits with oscillating terms fail to exist?

### Tip:
For oscillating functions like \( \sin\left(\frac{1}{x}\right) \), their behavior near 0 often requires careful analysis, as they can prevent limits from existing.

Consider the function f(x) = { 1/x^2 * sin(1/x), x ≠ 0; 0, x = 0 }. Then which of the following is not true?

Analyze the behavior of a piecewise function defined by 1/x^2 * sin(1/x) when x ≠ 0 and 0 when x = 0. Learn how to determine limits, assess continuity, and understand the impact of oscillatory functions on limit existence.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation