Math Problem Statement

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

Solution

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

\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.

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Math Problem Analysis

Mathematical Concepts

Limits
Continuity
Piecewise Functions
Oscillations

Formulas

lim (x → 0) [1/x^2 * sin(1/x)]
Continuity condition: lim (x → c) f(x) = f(c)

Theorems

Limit existence and oscillation behavior
Continuity of piecewise functions

Suitable Grade Level

Grades 11-12 or Early College