Math Problem Statement
Let f(x) = 4x - 3, if x < 2; (x - 3)^2, if x = 2; 3x + 1, if x > 2. Then find lim(x -> 2) f(x).
Solution
The question is asking to find the limit of the given piecewise function as . The function is defined as:
\begin{cases} 4x - 3, & x < 2 \\ (x - 3)^2, & x = 2 \\ 3x + 1, & x > 2 \end{cases}$$ We need to compute $$\lim_{x \to 2} f(x)$$, which involves finding the left-hand limit and the right-hand limit at $$x = 2$$ and then checking if they are equal. ### Step 1: Left-hand limit $$\lim_{x \to 2^-} f(x)$$ For $$x < 2$$, the function is given by $$4x - 3$$. Thus, we compute: $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (4x - 3) = 4(2) - 3 = 8 - 3 = 5$$ ### Step 2: Right-hand limit $$\lim_{x \to 2^+} f(x)$$ For $$x > 2$$, the function is given by $$3x + 1$$. Thus, we compute: $$\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (3x + 1) = 3(2) + 1 = 6 + 1 = 7$$ ### Step 3: Check if the limits are equal Since the left-hand limit is 5 and the right-hand limit is 7, the two limits are not equal. Therefore, the two-sided limit does not exist. Thus, the answer is **(a) Does not exist**. --- Would you like more details on any part of this solution? Here are some related questions to deepen your understanding: 1. How do we determine the limit for a piecewise function? 2. What does it mean when a two-sided limit does not exist? 3. Can a function still be continuous if the limit does not exist at a particular point? 4. How do left-hand and right-hand limits influence the overall limit of a function? 5. What role does the function definition at $$x = 2$$ play in finding the limit? **Tip**: Always verify if the left-hand limit equals the right-hand limit to confirm if the overall limit exists.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Formulas
Left-hand limit
Right-hand limit
Theorems
Limit Theorem
Piecewise Function Limits
Suitable Grade Level
Grades 11-12 (Calculus Introduction)