Math Problem Statement

Graph the following function and then find the specified limit. When​ necessary, state that the limit does not exist. ​G(x)equals left brace Start 2 By 2 Matrix 1st Row 1st Column negative x plus 2 comma 2nd Column for x less than 1 comma 2nd Row 1st Column x squared comma 2nd Column for x greater than 1. EndMatrix Find ModifyingBelow lim With x right arrow 1 Upper G left parenthesis x right parenthesis .

Solution

The given piecewise function is defined as:

\begin{cases} - x + 2 & \text{for} \, x < 1, \\ x^2 & \text{for} \, x > 1. \end{cases}$$ ### Step 1: Graphing the Function We need to graph the two parts of this piecewise function: 1. **For $$x < 1$$:** The function is $$G(x) = -x + 2$$, which is a linear function with slope -1 and intercept at $$(0, 2)$$. 2. **For $$x > 1$$:** The function is $$G(x) = x^2$$, which is a parabolic curve opening upwards. ### Step 2: Finding the Limit as $$x \to 1$$ The goal is to find: $$\lim_{x \to 1} G(x).$$ We need to check both the left-hand and right-hand limits at $$x = 1$$. - **Left-hand limit** ($$x \to 1^-$$): As $$x$$ approaches 1 from the left ($$x < 1$$), we use the piece $$G(x) = -x + 2$$. $$\lim_{x \to 1^-} G(x) = \lim_{x \to 1^-} (-x + 2) = -1 + 2 = 1.$$ - **Right-hand limit** ($$x \to 1^+$$): As $$x$$ approaches 1 from the right ($$x > 1$$), we use the piece $$G(x) = x^2$$. $$\lim_{x \to 1^+} G(x) = \lim_{x \to 1^+} (x^2) = 1^2 = 1.$$ ### Step 3: Conclusion on the Limit Since both the left-hand and right-hand limits are equal and equal to 1, we can conclude that: $$\lim_{x \to 1} G(x) = 1.$$ Let me know if you would like me to graph the function for you or if you have any questions! --- Here are 5 related questions to consider: 1. What happens to the limit if the function had a discontinuity at $$x = 1$$? 2. How would the limit change if the right-hand side of the piecewise function was $$G(x) = x^3$$ for $$x > 1$$? 3. Can you find the derivative of each piece of the function separately? 4. How would the limit behave if the function was defined differently at $$x = 1$$? 5. What does it mean when a limit does not exist? **Tip:** When analyzing piecewise functions, always check both left-hand and right-hand limits to determine whether a limit exists.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Limits
Linear Functions
Quadratic Functions

Formulas

Limit Definition
Linear Equation: y = mx + c
Quadratic Equation: y = ax^2 + bx + c

Theorems

Limit Theorem
Continuity

Suitable Grade Level

Grades 10-12

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