Math Problem Statement
Find the value of \lim_{x \to 0} f(x) where f(x) = \begin{cases} x & \text{if } x \neq 0, \\ 1 & \text{if } x = 0. \end{cases}
Solution
The problem asks to find:
\begin{cases} x & \text{if } x \neq 0, \\ 1 & \text{if } x = 0. \end{cases}$$ ### Step-by-step solution: We will evaluate the limit by considering the behavior of the function as $$x \to 0$$ from both sides (left and right). 1. **For $$x \neq 0$$**: The function is simply $$f(x) = x$$. Therefore, as $$x$$ approaches 0 (either from the left or right), $$f(x)$$ approaches 0. $$\lim_{x \to 0} f(x) = \lim_{x \to 0} x = 0$$ 2. **For $$x = 0$$**: At $$x = 0$$, the function is defined as $$f(0) = 1$$. ### Conclusion: The value of the function approaches 0 as $$x \to 0$$ for all $$x \neq 0$$. Therefore, the limit as $$x \to 0$$ does not depend on the specific value of $$f(0)$$. Hence, the limit is: $$\boxed{0}$$ Do you have any questions, or would you like further details? --- Here are 5 related questions that expand on this: 1. What is the difference between a limit and the actual value of a function at a point? 2. Can the limit of a function exist even if the function is not defined at the limit point? 3. How would the limit change if the function were defined as $$f(x) = 2x$$ for $$x \neq 0$$? 4. How do we handle limits that approach infinity instead of a finite point? 5. What is the significance of left-hand and right-hand limits when evaluating limits? ### Tip: Always check if the limit and the function's actual value match—this indicates continuity at that point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Continuity
Formulas
\lim_{x \to 0} f(x)
\lim_{x \to 0} x = 0
Theorems
Limit definition
Continuity at a point
Suitable Grade Level
Grades 11-12
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