Math Problem Statement

Is the function f(x) = { x^2 + 1, if x < 1; 3x, if x ≥ 1 } continuous at x = 1? Explain.

Solution

The problem asks if the function f(x)f(x) is continuous at x=1x = 1 and provides a piecewise function:

\begin{cases} x^2 + 1, & \text{if } x < 1 \\ 3x, & \text{if } x \geq 1 \end{cases}$$ To check the continuity of $$f(x)$$ at $$x = 1$$, we must verify the following conditions: 1. **$$f(1)$$ is defined**. 2. **The left-hand limit exists**: $$\lim_{x \to 1^-} f(x)$$. 3. **The right-hand limit exists**: $$\lim_{x \to 1^+} f(x)$$. 4. **The function value and the limits must be equal**: $$f(1) = \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x)$$. ### Step 1: Find $$f(1)$$ Since $$x = 1$$ falls under the second case (where $$x \geq 1$$): $$f(1) = 3(1) = 3$$ ### Step 2: Compute the left-hand limit ($$x \to 1^-$$) For $$x < 1$$, $$f(x) = x^2 + 1$$, so we find: $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x^2 + 1) = 1^2 + 1 = 2$$ ### Step 3: Compute the right-hand limit ($$x \to 1^+$$) For $$x \geq 1$$, $$f(x) = 3x$$, so: $$\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (3x) = 3(1) = 3$$ ### Step 4: Check continuity - $$f(1) = 3$$ - $$\lim_{x \to 1^-} f(x) = 2$$ - $$\lim_{x \to 1^+} f(x) = 3$$ Since $$\lim_{x \to 1^-} f(x) \neq \lim_{x \to 1^+} f(x)$$, the function is **not continuous** at $$x = 1$$. Let me know if you want more details or have any questions! --- Here are 5 related questions to deepen your understanding: 1. What are the conditions for a function to be continuous at a point? 2. How do piecewise functions affect continuity? 3. Can you give an example of a function that is continuous everywhere? 4. What happens when the left-hand and right-hand limits are not equal? 5. How would you define the concept of a limit in simple terms? **Tip:** When checking continuity at a point in piecewise functions, always carefully analyze each part of the function around that point.

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Math Problem Analysis

Mathematical Concepts

Continuity
Piecewise Functions
Limits

Formulas

Limit definition: lim_{x -> c} f(x)
Piecewise function conditions

Theorems

Continuity Definition
Limit Theorems

Suitable Grade Level

Grades 10-12