https://file.aiquickdraw.com/mathbot/file/172736343652594tfqz4q.jpg

Let's analyze the given problem.

The function is defined as:

\[
f(x) = 
\begin{cases} 
\frac{1}{x^3} \sin \left( \frac{1}{x^3} \right), & \text{if } x \neq 0 \\
0, & \text{if } x = 0 
\end{cases}
\]

We are asked to identify which statement is **not true** from the following options:

a. None of these  
b. The function is discontinuous at \( x = 0 \)  
c. The limit exists at \( x = 0 \)  
d. The limit does not exist at \( x = 0 \)

### Step-by-Step Analysis:

1. **Limit behavior as \( x \to 0 \):**

   - For \( x \neq 0 \), the function is \( \frac{1}{x^3} \sin \left( \frac{1}{x^3} \right) \). We know that \( \sin(\theta) \) is bounded between \(-1\) and \(1\), so:
     \[
     -\frac{1}{x^3} \leq \frac{1}{x^3} \sin \left( \frac{1}{x^3} \right) \leq \frac{1}{x^3}
     \]
     As \( x \to 0 \), the oscillations of \( \sin \left( \frac{1}{x^3} \right) \) become wilder, leading the function to oscillate more erratically without settling on a single value. Therefore, **the limit does not exist** as \( x \to 0 \).

2. **Discontinuity at \( x = 0 \):**

   - The function is given as 0 at \( x = 0 \). However, as shown in the previous step, the limit of the function as \( x \to 0 \) does not exist. Therefore, the function is **discontinuous at \( x = 0 \)**.

3. **Analyzing the options:**
   - **Option (b)**: The function is indeed discontinuous at \( x = 0 \). This statement is **true**.
   - **Option (c)**: The limit does not exist at \( x = 0 \). Therefore, this statement is **false**.
   - **Option (d)**: The limit does not exist, so this statement is **true**.
   - **Option (a)**: Since option (c) is false, this statement is also **false**.

### Conclusion:

The correct answer is **(c)**, as the statement "The limit exists at \( x = 0 \)" is **not true**.

Would you like further explanation on any part of the solution? Here are some related questions to deepen your understanding:

1. How can you formally prove that the limit does not exist for this function at \( x = 0 \)?
2. What are some common behaviors of oscillating functions as \( x \to 0 \)?
3. How do we test for discontinuity in functions like this one?
4. Why does the value of \( \sin \left( \frac{1}{x^3} \right) \) cause such erratic behavior as \( x \to 0 \)?
5. What are other examples of functions where the limit does not exist at a particular point due to oscillation?

**Tip:** Always check both one-sided limits when determining whether a limit exists at a point. If the one-sided limits differ, the overall limit does not exist.

Consider the function f(x) = {1/x^3 * sin(1/x^3), x ≠ 0; 0, x = 0}. Then which of the following is not true?

Explore the function f(x) = 1/x^3 * sin(1/x^3) and its behavior at x = 0. This problem delves into limits, continuity, and the concept of oscillatory functions. Learn how to determine if a limit exists and identify discontinuity at a point.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation