Math Problem Statement
Consider the function f(x) defined as a piecewise function. Determine the values of a for which the limit lim(x→a) f(x) exists.
Solution
To determine the values of for which the limit exists, we need to analyze the behavior of the piecewise function at the points where the function definition changes. The function is defined as:
\begin{cases} 2 - x & \text{if } x < -1 \\ x & \text{if } -1 \leq x < 2 \\ (x - 2)^2 & \text{if } x \geq 2 \end{cases}$$ ### Steps to Determine $$\lim_{x \to a} f(x)$$ Existence: 1. **Check Continuity at $$a = -1$$:** - Left-hand limit as $$x$$ approaches $$-1$$ from the left: $$\lim_{x \to -1^-} f(x) = \lim_{x \to -1^-} (2 - x) = 2 - (-1) = 3$$ - Right-hand limit as $$x$$ approaches $$-1$$ from the right: $$\lim_{x \to -1^+} f(x) = \lim_{x \to -1^+} x = -1$$ - Since $$3 \neq -1$$, the limit does **not** exist at $$a = -1$$. 2. **Check Continuity at $$a = 2$$:** - Left-hand limit as $$x$$ approaches $$2$$ from the left: $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} x = 2$$ - Right-hand limit as $$x$$ approaches $$2$$ from the right: $$\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (x - 2)^2 = (2 - 2)^2 = 0$$ - Since $$2 \neq 0$$, the limit does **not** exist at $$a = 2$$. ### Conclusion: The limit $$\lim_{x \to a} f(x)$$ exists for all values of $$a$$ **except** $$a = -1$$ and $$a = 2$$. --- Do you want further details or have any questions? ### Related Questions: 1. What is the definition of a piecewise function? 2. How do you determine if a function is continuous at a point? 3. Why is the concept of limits important in calculus? 4. How can one visualize the behavior of limits using a graph? 5. What are one-sided limits and how are they used in continuity tests? ### Tip: When analyzing the existence of limits for piecewise functions, always check the function at the points where the piecewise conditions change, as these are potential points of discontinuity.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Continuity
Formulas
lim(x→a) f(x)
One-sided limits
Theorems
Continuity Test
Limit Existence Theorem
Suitable Grade Level
Grades 11-12 (Pre-Calculus/Calculus Level)