Math Problem Statement
١ متصل عند س= ٢ ٢ متصل والنهاية غير موجودة عند س= ٢ ٣ غير متصل والنهاية موجودة عند س=٢وخيار رقم اربعة هو غير منصل والنهاية غير موجودة عند س = ٢
Solution
To solve this, we need to determine the continuity and behavior of the function around , given the provided piecewise function.
Analysis:
The function is defined as:
\begin{cases} \frac{x^2 - 4}{x - 2}, & x \neq 2 \\ 2, & x = 2 \end{cases}$$ 1. **Simplify $$f(x)$$ for $$x \neq 2$$:** $$f(x) = \frac{x^2 - 4}{x - 2} = \frac{(x - 2)(x + 2)}{x - 2}, \quad x \neq 2$$ Cancelling $$x - 2$$ (valid since $$x \neq 2$$): $$f(x) = x + 2, \quad x \neq 2$$ So, for $$x \neq 2$$, the function behaves like $$x + 2$$. 2. **Find the limit as $$x \to 2$$:** Using the simplified expression $$f(x) = x + 2$$ for $$x \neq 2$$: $$\lim_{x \to 2} f(x) = \lim_{x \to 2} (x + 2) = 4$$ 3. **Compare the limit to $$f(2)$$:** By definition: $$f(2) = 2$$ Since $$\lim_{x \to 2} f(x) = 4 \neq f(2) = 2$$, the function is **not continuous** at $$x = 2$$. 4. **Determine if the limit exists:** The limit as $$x \to 2$$ exists and is equal to 4. However, this does not match the value of $$f(2)$$. ### Conclusion: The correct choice is **(3)**: The function is not continuous at $$x = 2$$, but the limit exists at $$x = 2$$. Let me know if you would like further clarification or a detailed breakdown of any step! --- **Follow-Up Questions:** 1. What is the definition of continuity for a function at a point? 2. Why do we need to simplify $$f(x)$$ for $$x \neq 2$$? 3. How does canceling $$x - 2$$ affect the domain of the function? 4. What happens if the limit does not exist at $$x = 2$$? 5. How would the piecewise definition of $$f(x)$$ change the continuity behavior? **Tip:** Always check the limit and function value separately to confirm continuity at a point!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity at a Point
Limits
Formulas
Simplification of \( \frac{x^2 - 4}{x - 2} \) to \( x + 2 \) for \( x \neq 2 \)
Definition of Continuity: \( \lim_{x \to c} f(x) = f(c) \)
Theorems
Limits and Continuity Theorem
Suitable Grade Level
Grade 10-12
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