Math Problem Statement
Q5 - Explain why or why not the function is continuous at a = 2. a. Use Math Notation b. Provide an English sentence. f(x) = {1 - x^2 if 0 ≤ x < 2; 2x + 1 if x ≥ 2}
Solution
Let's analyze the function and determine whether it is continuous at .
The function is given as:
1 - x^2 & \text{if } 0 \leq x < 2 \\ 2x + 1 & \text{if } x \geq 2 \end{cases}$$ To check continuity at $$x = 2$$, we need to verify the following conditions: 1. **Existence of $$f(2)$$**: $$f(2) = 2(2) + 1 = 5$$ 2. **Existence of the left-hand limit as $$x \to 2^-$$** (i.e., approaching 2 from the left): $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (1 - x^2) = 1 - 2^2 = 1 - 4 = -3$$ 3. **Existence of the right-hand limit as $$x \to 2^+$$** (i.e., approaching 2 from the right): $$\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (2x + 1) = 2(2) + 1 = 5$$ 4. **Continuity condition**: A function is continuous at $$x = 2$$ if: $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = f(2)$$ Here, we observe that: $$\lim_{x \to 2^-} f(x) = -3 \quad \text{and} \quad \lim_{x \to 2^+} f(x) = 5$$ Since $$-3 \neq 5$$, the left-hand and right-hand limits are not equal. Therefore, the function is **not continuous** at $$x = 2$$. ### English Explanation: The function $$f(x)$$ is not continuous at $$x = 2$$ because the left-hand limit as $$x$$ approaches 2 is $$-3$$, while the right-hand limit is 5. Since these two limits are not equal, and the value of the function at $$x = 2$$ is 5, the function has a discontinuity at this point. --- Would you like more details or have any questions? Here are 5 related questions for further understanding: 1. What is the definition of a continuous function? 2. How do we compute one-sided limits for piecewise functions? 3. Why does the value of $$f(2)$$ not influence the left-hand limit? 4. How can we graphically represent discontinuities in piecewise functions? 5. What types of discontinuities can a function have? **Tip**: Always verify both one-sided limits when checking for continuity at a point in piecewise functions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
f(2) = 2(2) + 1 = 5
lim_{x -> 2^-} (1 - x^2) = -3
lim_{x -> 2^+} (2x + 1) = 5
Theorems
Definition of Continuity
Limit Theorem
Suitable Grade Level
Grades 11-12