Math Problem Statement
Find the limit if it exists: \(\lim_{{x \to 0}} f(x)\), where \(f(x) = \begin{cases} -1, & x < 0 \\ 1, & x > 0 \end{cases}\).
Solution
The image contains a mathematical problem asking to find the limit of a piecewise function as . The function is defined as follows:
-1, & x < 0 \\ 1, & x > 0 \end{cases}$$ The problem asks to evaluate the limit: $$\lim_{x \to 0} f(x)$$ ### Solution: To evaluate the limit, we must check the left-hand limit (as $$x$$ approaches 0 from the left) and the right-hand limit (as $$x$$ approaches 0 from the right). 1. **Left-hand limit ($$x \to 0^-$$):** When $$x \to 0^-$$, meaning $$x$$ approaches 0 from the left (negative side), the function takes the value $$f(x) = -1$$. $$\lim_{x \to 0^-} f(x) = -1$$ 2. **Right-hand limit ($$x \to 0^+$$):** When $$x \to 0^+$$, meaning $$x$$ approaches 0 from the right (positive side), the function takes the value $$f(x) = 1$$. $$\lim_{x \to 0^+} f(x) = 1$$ ### Conclusion: Since the left-hand limit $$-1$$ and the right-hand limit $$1$$ are not equal, the two-sided limit does not exist. Therefore, the limit does not exist: $$\lim_{x \to 0} f(x) \text{ does not exist}.$$ Would you like more details or explanations? Here are some related questions you might find helpful: 1. What are the conditions for a limit to exist? 2. How do piecewise functions affect limits? 3. Can limits exist if the left and right limits are not equal? 4. What is the difference between one-sided and two-sided limits? 5. How are discontinuities related to limits? **Tip:** When analyzing limits of piecewise functions, always check both the left-hand and right-hand limits to determine if the overall limit exists.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Calculus
Limits
Piecewise Functions
Formulas
\(\lim_{{x \to a}} f(x)\)
Left-hand limit \(\lim_{{x \to a^-}} f(x)\)
Right-hand limit \(\lim_{{x \to a^+}} f(x)\)
Theorems
Limit does not exist if left-hand limit and right-hand limit are not equal
Suitable Grade Level
Grades 11-12 or early college