Math Problem Statement
What is the limit \(\lim_{x \to 3} f(x)\) for the given piecewise function: \( f(x) = \begin{cases} 2x + 1 & \text{if } x < 3 \\ 7 & \text{if } x = 3 \\ x + 4 & \text{if } x > 3 \end{cases} \)?
Solution
We are tasked with finding the limit for the given piecewise function:
2x + 1 & \text{if } x < 3 \\ 7 & \text{if } x = 3 \\ x + 4 & \text{if } x > 3 \end{cases}$$ To evaluate this limit, we must check the left-hand limit $$\lim_{x \to 3^-} f(x)$$ and the right-hand limit $$\lim_{x \to 3^+} f(x)$$. 1. **Left-hand limit** ($$x \to 3^-$$): - For $$x < 3$$, $$f(x) = 2x + 1$$. - Substituting $$x = 3$$ into this expression: $$\lim_{x \to 3^-} f(x) = 2(3) + 1 = 6 + 1 = 7.$$ 2. **Right-hand limit** ($$x \to 3^+$$): - For $$x > 3$$, $$f(x) = x + 4$$. - Substituting $$x = 3$$ into this expression: $$\lim_{x \to 3^+} f(x) = 3 + 4 = 7.$$ Since both the left-hand limit and right-hand limit are equal to 7, we conclude that: $$\lim_{x \to 3} f(x) = 7.$$ Thus, the correct answer is **A) 7**. Would you like more details, or do you have any questions? Here are five related questions: 1. How would the limit change if the right-hand limit expression was different? 2. What if the piecewise function had a discontinuity at $$x = 3$$? 3. How would you define continuity based on this example? 4. What happens if you change the middle value $$f(3)$$ from 7 to something else? 5. Can you explain how the piecewise function affects limits in general? **Tip:** Always check both the left-hand and right-hand limits when dealing with piecewise functions to ensure they match for the limit to exist.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Formulas
Limit definition: \( \lim_{x \to a} f(x) = L \)
Evaluating limits using left-hand and right-hand limits
Theorems
Limit definition
Continuity theorem
Suitable Grade Level
Grades 10-12