Math Problem Statement
If f is the function defined above, then \(\lim_{{x \to 2}} f(x)\) is:
Solution
To solve the problem of finding where the function is defined as:
\frac{(x - 2)^2 \sin x}{|x - 2|} & \text{for } x \neq 2 \\ \pi & \text{for } x = 2 \end{cases}$$ ### Step 1: Analyze the limit for $$x \neq 2$$ For $$x \neq 2$$, the function is: $$f(x) = \frac{(x - 2)^2 \sin x}{|x - 2|}$$ This can be simplified to: $$f(x) = |x - 2| \cdot (x - 2) \cdot \sin x$$ ### Step 2: Take the limit as $$x$$ approaches 2 We are interested in: $$\lim_{{x \to 2}} f(x) = \lim_{{x \to 2}} |x - 2| \cdot (x - 2) \cdot \sin x$$ ### Step 3: Evaluate the components of the limit 1. $$|x - 2|$$ approaches 0 as $$x$$ approaches 2. 2. $$(x - 2)$$ also approaches 0 as $$x$$ approaches 2. 3. $$\sin x$$ approaches $$\sin 2$$, which is a finite number. ### Step 4: Conclusion Since both $$|x - 2|$$ and $$(x - 2)$$ approach 0, the entire product approaches 0. Therefore: $$\lim_{{x \to 2}} f(x) = 0$$ ### Step 5: Consider the given function at $$x = 2$$ The function is defined as $$f(2) = \pi$$. However, the limit depends on the function's behavior as $$x$$ approaches 2 from both sides, not just the value at $$x = 2$$. Since the limit is determined by the function's behavior for $$x \neq 2$$, we conclude: $$\lim_{{x \to 2}} f(x) = 0$$ ### Answer The correct answer is $$\boxed{0}$$ (Option A). Would you like further details or explanations? Here are five questions related to this topic: 1. What is the difference between a limit and the actual value of a function at a point? 2. How does the concept of continuity relate to limits? 3. What would happen if the limit did not exist at $$x = 2$$? 4. How do we determine limits for piecewise functions in general? 5. Can a limit exist even if the function is undefined at that point? **Tip:** When approaching limit problems, always consider simplifying the expression before directly substituting the value to avoid indeterminate forms.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Trigonometry
Formulas
\(\lim_{{x \to 2}} f(x)\)
\(f(x) = \frac{(x - 2)^2 \sin x}{|x - 2|}\) for \(x \neq 2\)
Theorems
Limit Theorem
Squeeze Theorem
Suitable Grade Level
Grades 11-12