Math Problem Statement
Let f be the function defined as f(x) = (x^2 - 4) / (x - 2) if x ≠ 2 and f(x) = 1 if x = 2. Which of the following statements are true? I. f has a limit at x = 2, II. f is continuous at x = 2, III. f is differentiable at x = 2.
Solution
Let's analyze the given function :
\begin{cases} \frac{x^2 - 4}{x - 2} & \text{if } x \neq 2 \\ 1 & \text{if } x = 2 \end{cases}$$ We need to determine which statements about $$f$$ are true: ### I. $$f$$ has a limit at $$x = 2$$ To check the limit as $$x$$ approaches 2, we simplify the expression for $$f(x)$$ when $$x \neq 2$$. $$f(x) = \frac{x^2 - 4}{x - 2} = \frac{(x - 2)(x + 2)}{x - 2} = x + 2 \quad \text{for } x \neq 2.$$ Thus, for $$x \neq 2$$, $$f(x) = x + 2$$. Now, we check the limit as $$x$$ approaches 2: $$\lim_{x \to 2} f(x) = \lim_{x \to 2} (x + 2) = 4.$$ Since $$f(2) = 1$$, the value of the function at $$x = 2$$ is not equal to the limit as $$x$$ approaches 2. However, **the limit exists** and equals 4, so statement I is **true**. ### II. $$f$$ is continuous at $$x = 2$$ For continuity at $$x = 2$$, the following must hold: 1. $$\lim_{x \to 2} f(x)$$ exists (which we found to be 4), 2. $$f(2)$$ exists (we know $$f(2) = 1$$), 3. $$\lim_{x \to 2} f(x) = f(2)$$. Since $$\lim_{x \to 2} f(x) = 4$$ and $$f(2) = 1$$, the function is **not continuous** at $$x = 2$$. Therefore, statement II is **false**. ### III. $$f$$ is differentiable at $$x = 2$$ For differentiability, the function must be continuous at $$x = 2$$. Since we just established that $$f$$ is not continuous at $$x = 2$$, it cannot be differentiable at that point. Thus, statement III is **false**. ### Conclusion: The only true statement is I, so the correct answer is **A) I only**. --- Would you like more details on any part of the solution? Here are five related questions you might find helpful: 1. What is the definition of continuity, and how is it related to limits? 2. Can a function have a limit but still be discontinuous at a point? 3. What is the formal definition of differentiability, and how does it relate to continuity? 4. Can a function be differentiable but not continuous? Why or why not? 5. How do you simplify rational functions like $$\frac{x^2 - 4}{x - 2}$$ for limits? **Tip**: Always check for removable discontinuities by simplifying the function before finding limits or continuity.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Continuity
Differentiability
Formulas
Limit definition: lim_{x -> a} f(x)
Factoring polynomials: x^2 - 4 = (x - 2)(x + 2)
Theorems
Limit definition
Conditions for continuity
Conditions for differentiability
Suitable Grade Level
Grades 11-12 (Precalculus/Calculus level)
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