Math Problem Statement
Estimate \( \lim_{x \to 4} f(x) \) where the piecewise function is defined as: \( f(x) = x - 2 \) for \( x < 4 \), and \( f(x) = x + 2 \) for \( x \geq 4 \).
Solution
To solve this problem, let's break it down step by step:
Problem:
We are tasked with estimating the limit of as . The piecewise function is defined as:
\begin{cases} x - 2 & \text{if } x < 4, \\ x + 2 & \text{if } x \geq 4. \end{cases}$$ The graph of the function is also provided, and we are given four possible answers: $$8$$, $$6$$, $$2$$, or "The limit does not exist." --- ### Step 1: Analyze the left-hand limit ($$\lim_{x \to 4^-} f(x)$$) From the piecewise definition: $$f(x) = x - 2 \quad \text{for } x < 4.$$ Substituting $$x = 4$$ in this expression to find the left-hand limit: $$\lim_{x \to 4^-} f(x) = 4 - 2 = 2.$$ --- ### Step 2: Analyze the right-hand limit ($$\lim_{x \to 4^+} f(x)$$) From the piecewise definition: $$f(x) = x + 2 \quad \text{for } x \geq 4.$$ Substituting $$x = 4$$ in this expression to find the right-hand limit: $$\lim_{x \to 4^+} f(x) = 4 + 2 = 6.$$ --- ### Step 3: Compare the left-hand and right-hand limits For the overall limit $$\lim_{x \to 4} f(x)$$ to exist, the left-hand limit and right-hand limit must be equal. However: $$\lim_{x \to 4^-} f(x) = 2 \quad \text{and} \quad \lim_{x \to 4^+} f(x) = 6.$$ Since these two values are not equal, the limit does **not exist**. --- ### Final Answer: The correct choice is: **"Limit does not exist."** --- Would you like more details or further explanation? Here are 5 related questions to deepen your understanding: 1. What does it mean for a limit to exist at a point? 2. How can you analyze limits graphically versus algebraically? 3. How are left-hand and right-hand limits used to determine the existence of a limit? 4. Can a function still be continuous if the limit at a point does not exist? 5. What are some real-world examples of piecewise functions and their applications? **Tip:** Always compare left-hand and right-hand limits when evaluating the limit at a point, especially for piecewise functions!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Continuity
Formulas
Left-hand limit (\( \lim_{x \to c^-} \))
Right-hand limit (\( \lim_{x \to c^+} \))
Theorems
Limit Existence Rule: A limit exists at a point only if the left-hand and right-hand limits are equal.
Suitable Grade Level
Grades 10-12