Math Problem Statement
Given the following piecewise function, evaluate lim x→1⁻ f(x).
Solution
We are asked to evaluate the limit:
where is a piecewise function defined as:
-2x^2 - 2x & \text{if } x \leq -2 \\ 3x - 2 & \text{if } -2 < x < 1 \\ x^2 + 3 & \text{if } x > 1 \end{cases}$$ We are particularly interested in the left-hand limit as $$x \to 1^-$$. In this case, $$-2 < x < 1$$, so we use the second piece of the function: $$f(x) = 3x - 2$$. To find the limit: $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (3x - 2) = 3(1) - 2 = 1$$ Thus, the value of the limit is $$\boxed{1}$$. Let me know if you'd like further details or have questions. Here are five related questions: 1. How do you compute a right-hand limit for the same function? 2. What happens when $$x \to -2^+$$? 3. How does a discontinuity affect the value of a limit? 4. How can piecewise functions have different behaviors at boundaries? 5. Can a limit exist if the left-hand and right-hand limits are different? **Tip:** Always check which part of a piecewise function applies based on the value you're approaching.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Calculus
Formulas
lim x→c f(x)
f(x) = { -2x^2 - 2x for x ≤ -2; 3x - 2 for -2 < x < 1; x^2 + 3 for x > 1 }
Theorems
Limit Definition for Piecewise Functions
Left-Hand Limit Calculation
Suitable Grade Level
High School - Calculus (Grades 11-12)