Math Problem Statement
Determine if the function g(x), defined as piecewise:
g(x) =
{ x^2 + 2 for x < 2
{ 3x - 1 for x ≥ 2
is continuous at x = 2, and explain why.
Solution
The problem in your image concerns checking the continuity of the piecewise function at . Let me walk you through the key aspects:
Given Function
The function is defined as:
\begin{cases} x^2 + 2 & \text{if } x < 2, \\ 3x - 1 & \text{if } x \geq 2. \end{cases}$$ ### Continuity at $$x_0 = 2$$ For $$g(x)$$ to be continuous at $$x_0 = 2$$, the following condition must hold: $$\lim_{x \to 2^-} g(x) = \lim_{x \to 2^+} g(x) = g(2).$$ 1. **Left-hand limit** $$\lim_{x \to 2^-} g(x)$$: Since $$x < 2$$ corresponds to $$g(x) = x^2 + 2$$, $$\lim_{x \to 2^-} g(x) = (2)^2 + 2 = 6.$$ 2. **Right-hand limit** $$\lim_{x \to 2^+} g(x)$$: Since $$x \geq 2$$ corresponds to $$g(x) = 3x - 1$$, $$\lim_{x \to 2^+} g(x) = 3(2) - 1 = 5.$$ 3. **Conclusion**: The left-hand limit and the right-hand limit do not coincide: $$\lim_{x \to 2^-} g(x) = 6 \quad \text{and} \quad \lim_{x \to 2^+} g(x) = 5.$$ Therefore, $$g(x)$$ is **not continuous** at $$x_0 = 2$$. --- ### Final Answer The function $$g(x)$$ has a **discontinuity** at $$x_0 = 2$$ because the left-hand and right-hand limits are different. --- Do you want further clarifications on the definitions of continuity or limits? Here are 5 related questions to explore further: 1. How can a function have a jump discontinuity, and what does it mean visually? 2. What are the conditions for a piecewise function to be continuous everywhere? 3. How can we classify discontinuities into removable, jump, and infinite types? 4. If $$g(x)$$ were modified, how could it be made continuous at $$x_0 = 2$$? 5. How do left-hand and right-hand derivatives relate to discontinuities in piecewise functions? **Tip**: Always check both the left-hand limit and right-hand limit when testing continuity at a specific point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity of functions
Limits
Piecewise functions
Formulas
Left-hand limit: lim x→a⁻ f(x)
Right-hand limit: lim x→a⁺ f(x)
Continuity condition: lim x→a⁻ f(x) = lim x→a⁺ f(x) = f(a)
Theorems
Definition of continuity at a point
Properties of limits for piecewise functions
Suitable Grade Level
High School (Grades 10-12)