Math Problem Statement
Xét tính liên tục của hàm số f(x): f(x) = (3x^2 + |x|)/x, nếu x ≠ 0 và f(x) = 1, nếu x = 0.
Solution
The problem is asking to check the continuity of the given function , which is defined as follows:
\frac{3x^2 + |x|}{x}, & \text{if } x \neq 0, \\ 1, & \text{if } x = 0. \end{cases}$$ To check the continuity of this function at $$x = 0$$, we need to ensure that the following conditions hold: 1. The function is defined at $$x = 0$$, which it is, since $$f(0) = 1$$. 2. The limit of $$f(x)$$ as $$x \to 0$$ exists. 3. The limit of $$f(x)$$ as $$x \to 0$$ must be equal to $$f(0)$$. ### Step 1: Simplify $$f(x)$$ for $$x \neq 0$$ For $$x \neq 0$$: $$f(x) = \frac{3x^2 + |x|}{x}.$$ We can break it down into two cases for $$|x|$$: - When $$x > 0$$, $$|x| = x$$, so: $$f(x) = \frac{3x^2 + x}{x} = 3x + 1.$$ - When $$x < 0$$, $$|x| = -x$$, so: $$f(x) = \frac{3x^2 - x}{x} = 3x - 1.$$ Thus, the function $$f(x)$$ becomes: $$f(x) = \begin{cases} 3x + 1, & \text{if } x > 0, \\ 3x - 1, & \text{if } x < 0. \end{cases}$$ ### Step 2: Calculate the limit of $$f(x)$$ as $$x \to 0$$ - For $$x \to 0^+$$ (from the right): $$\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (3x + 1) = 1.$$ - For $$x \to 0^-$$ (from the left): $$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (3x - 1) = -1.$$ ### Step 3: Check the continuity at $$x = 0$$ The left-hand limit as $$x \to 0^-$$ is $$-1$$, and the right-hand limit as $$x \to 0^+$$ is $$1$$. Since these two limits are not equal, the limit of $$f(x)$$ as $$x \to 0$$ does not exist. Therefore, $$f(x)$$ is **discontinuous** at $$x = 0$$, because the second condition (the existence of the limit) is not satisfied. --- Would you like more details, or do you have any other questions? Here are 5 related questions: 1. What are the different types of discontinuities a function can have? 2. How do you prove the continuity of a piecewise function at a given point? 3. Can a function be continuous everywhere except at a single point? What is an example? 4. How does the absolute value function affect continuity in piecewise functions? 5. How does the derivative of a piecewise function behave at discontinuous points? **Tip**: For checking continuity, always verify the existence of the limit from both directions and ensure it matches the function value at the point in question.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Piecewise Functions
Absolute Value
Formulas
f(x) = (3x^2 + |x|)/x, nếu x ≠ 0
f(x) = 1, nếu x = 0
Lim_{x -> 0^+} f(x)
Lim_{x -> 0^-} f(x)
Theorems
Definition of continuity: A function is continuous at x = a if f(a) is defined, the limit of f(x) as x approaches a exists, and Lim_{x -> a} f(x) = f(a).
Limit properties for piecewise functions
Suitable Grade Level
High School (Grades 10-12)