Math Problem Statement
Given the function f(x) = {2 - x^2 for x ≤ 3, 21/x for x > 3}, find lim(x→3) f(x)
Solution
We are tasked with finding the limit of the given function as approaches 3. The function is defined piecewise as follows:
\begin{cases} 2 - x^2 & \text{if } x \leq 3, \\ \frac{21}{x} & \text{if } x > 3. \end{cases}$$ ### Step 1: Evaluate the Left-Hand Limit ($$\lim_{x \to 3^-} f(x)$$) For $$x \leq 3$$, the function is $$f(x) = 2 - x^2$$. $$\lim_{x \to 3^-} f(x) = 2 - 3^2 = 2 - 9 = -7.$$ ### Step 2: Evaluate the Right-Hand Limit ($$\lim_{x \to 3^+} f(x)$$) For $$x > 3$$, the function is $$f(x) = \frac{21}{x}$$. $$\lim_{x \to 3^+} f(x) = \frac{21}{3} = 7.$$ ### Step 3: Determine if the Limit Exists Since the left-hand limit ($$\lim_{x \to 3^-} f(x) = -7$$) and the right-hand limit ($$\lim_{x \to 3^+} f(x) = 7$$) are not equal, the two-sided limit does not exist. Thus, the correct answer is: $$\boxed{D. \text{The limit does not exist}}.$$ Let me know if you would like further details or have any questions. Here are five related questions for further exploration: 1. What is the definition of a limit for piecewise functions? 2. How do you determine continuity at a point for piecewise functions? 3. How would the limit change if the functions on both sides of the piecewise definition were equal at $$x = 3$$? 4. What does it mean for a function to have a one-sided limit but no two-sided limit? 5. How does the concept of limits relate to derivatives? **Tip:** Always check both the left-hand and right-hand limits when dealing with piecewise functions to determine if the overall limit exists.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Formulas
lim(x→a) f(x)
Left-hand limit and Right-hand limit
Theorems
Limit of a piecewise function
Suitable Grade Level
Grades 10-12