Math Problem Statement
Graph the function. Determine the limit (if it exists) by evaluating the corresponding one-sided limits. The function is defined as f(x) = {x - 1, x ≤ 2; 2x - 3, x > 2} and we need to find the limit as x approaches 2.
Solution
The problem asks to evaluate the limit of a piecewise function as approaches 2, and determine whether it exists by evaluating the one-sided limits.
The function is defined as:
x - 1 & \text{for } x \leq 2 \\ 2x - 3 & \text{for } x > 2 \end{cases}$$ ### Steps to solve: 1. **Left-hand limit**: We evaluate the limit as $$x \to 2^-$$ (approaching 2 from the left). For $$x \leq 2$$, the function is $$f(x) = x - 1$$. $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (x - 1) = 2 - 1 = 1$$ 2. **Right-hand limit**: We evaluate the limit as $$x \to 2^+$$ (approaching 2 from the right). For $$x > 2$$, the function is $$f(x) = 2x - 3$$. $$\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (2x - 3) = 2(2) - 3 = 4 - 3 = 1$$ 3. **Comparing the limits**: Since both the left-hand limit and right-hand limit are equal: $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = 1$$ The two-sided limit exists and is equal to 1: $$\lim_{x \to 2} f(x) = 1$$ ### Graph of the Function: The function has two different expressions depending on whether $$x$$ is less than, equal to, or greater than 2. - For $$x \leq 2$$, the function is $$f(x) = x - 1$$, which is a line with a slope of 1. - For $$x > 2$$, the function is $$f(x) = 2x - 3$$, which is a line with a slope of 2. The graph should show continuity at $$x = 2$$, with both pieces meeting at the point $$(2, 1)$$. Would you like a more detailed explanation or assistance with graphing this? Here are 5 related questions to expand on this topic: 1. What happens if the left-hand and right-hand limits are not equal at a point? 2. How would you define continuity of a function at a point in terms of limits? 3. What is the difference between one-sided and two-sided limits? 4. How do you graph piecewise functions in general? 5. Can a piecewise function have a jump discontinuity, and how can you identify it? **Tip**: When evaluating limits, always check both one-sided limits if the function is piecewise-defined at the point in question.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
One-Sided Limits
Continuity
Formulas
Left-hand limit: lim_{x -> 2^-} f(x)
Right-hand limit: lim_{x -> 2^+} f(x)
Theorems
Limit Existence Theorem: A limit exists if and only if the left-hand limit equals the right-hand limit.
Suitable Grade Level
Grades 10-12