Math Problem Statement
Explain why f(x) = { x^2 if x ≠ 0, 1 if x = 0 } is not continuous at x = 0. Options: a) lim f(x) ≠ f(0), b) lim f(x) = f(0), c) lim f(x) ≠ f(1), d) lim f(x) does not exist.
Solution
Let's break down the problem and explain why the function is not continuous at .
The function given is:
\begin{cases} x^2 & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$ ### Continuity at $$x = 0$$: For a function to be continuous at a point $$x = c$$, it must satisfy three conditions: 1. **$$f(c)$$ is defined**. 2. **$$\lim_{x \to c} f(x)$$ exists**. 3. **$$\lim_{x \to c} f(x) = f(c)$$**. Let’s apply these conditions to $$f(x)$$ at $$x = 0$$: - **Condition 1**: $$f(0)$$ is defined. From the definition of $$f(x)$$, we know $$f(0) = 1$$. So, this condition is satisfied. - **Condition 2**: $$\lim_{x \to 0} f(x)$$ exists. Since for $$x \neq 0$$, $$f(x) = x^2$$, we need to check the limit of $$f(x)$$ as $$x \to 0$$. $$\lim_{x \to 0} f(x) = \lim_{x \to 0} x^2 = 0$$ The limit exists and equals 0. - **Condition 3**: $$\lim_{x \to 0} f(x) = f(0)$$. Here, $$\lim_{x \to 0} f(x) = 0$$, but $$f(0) = 1$$. Since 0 is not equal to 1, this condition is **not satisfied**. Thus, $$f(x)$$ is not continuous at $$x = 0$$ because the limit of $$f(x)$$ as $$x \to 0$$ does not equal $$f(0)$$. ### Correct Answer: From the given choices, the correct explanation is: - **a.** $$\lim_{x \to 0} f(x) \neq f(0)$$ Would you like more details on any step or further clarification? Here are some related questions to consider: 1. What is the definition of a discontinuous function? 2. How do you calculate the left-hand and right-hand limits? 3. Can a function be discontinuous at multiple points? 4. What happens to a function’s limit if it oscillates near a point? 5. How does the type of discontinuity affect the graph of a function? **Tip:** When analyzing continuity, always check both the limit and the function value at the given point!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Continuity
Piecewise Function
Formulas
lim_{x→0} x^2
f(x) = 1 if x = 0
Theorems
Definition of Continuity: lim_{x→c} f(x) = f(c)
Limit Theorem
Suitable Grade Level
Grades 10-12