Math Problem Statement
Cho hàm số f(x) = {sin(x)/|x| nếu x ≠ 0; 1 nếu x = 0}. Chọn khẳng định đúng về tính liên tục của f(x).
Solution
We are given a piecewise function defined as follows:
\begin{cases} \frac{\sin x}{|x|}, & \text{if } x \neq 0, \\ 1, & \text{if } x = 0. \end{cases}$$ The task is to determine which statement about the continuity of the function $$f(x)$$ is correct. ### Continuity at $$x = 0$$: To check if the function is continuous at $$x = 0$$, we need to verify if: $$\lim_{x \to 0} f(x) = f(0).$$ - **Step 1: Calculate the limit of $$f(x)$$ as $$x \to 0$$.** For $$x \neq 0$$, $$f(x) = \frac{\sin x}{|x|}$$. We examine the limit from both sides: - For $$x \to 0^+$$ (approaching from the right), $$f(x) = \frac{\sin x}{x} \quad \text{and} \quad \lim_{x \to 0^+} \frac{\sin x}{x} = 1.$$ - For $$x \to 0^-$$ (approaching from the left), $$f(x) = \frac{\sin x}{-x} \quad \text{and} \quad \lim_{x \to 0^-} \frac{\sin x}{-x} = -1.$$ - **Step 2: Compare the limits.** Since the left-hand limit is $$-1$$ and the right-hand limit is $$1$$, the two-sided limit does not exist. Therefore, the limit $$\lim_{x \to 0} f(x)$$ does not exist. - **Step 3: Compare with $$f(0)$$.** We are given that $$f(0) = 1$$. Since the limit does not exist, the function is **not continuous** at $$x = 0$$. ### Continuity at $$x = \pi$$: For $$x = \pi \neq 0$$, the function is $$f(\pi) = \frac{\sin \pi}{|\pi|} = 0$$, and this value is continuous at $$x = \pi$$ because the limit exists and matches the function value. ### Conclusion: - $$f(x)$$ is **not continuous** at $$x = 0$$. - The correct answer is option **D**: "f không liên tục phải tại $$x = 0$$" (f is not continuous at $$x = 0$$). Would you like more details, or have any other questions? Here are five follow-up questions for deeper understanding: 1. What is the general definition of continuity for a function at a point? 2. How do one-sided limits contribute to determining continuity at a point? 3. What other types of discontinuities can occur besides this type? 4. Why does the function behave differently at $$x = 0$$ compared to $$x = \pi$$? 5. Can the definition of this function be modified to make it continuous at $$x = 0$$? **Tip**: Always check the behavior of a function by evaluating both the left-hand and right-hand limits for key points where discontinuity might occur.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity of Piecewise Functions
Limits
One-Sided Limits
Formulas
f(x) = sin(x)/|x|, x ≠ 0
f(x) = 1, x = 0
lim(x -> 0) f(x)
Theorems
Definition of Continuity at a Point
Limit Theorem for One-Sided Limits
Suitable Grade Level
Grade 12 or College Level
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