Math Problem Statement
Consider the function f(x) defined as: f(x) = {e^x * sin(2πx), x < 0; x^2 - cx, x >= 0}. Choose the option that describes all values of c such that f(x) is continuous at x = 0.
Solution
To determine the value of that makes the function continuous at , we need to ensure that the limit of from both sides of is equal and that is well-defined.
The function is defined as:
\begin{cases} e^x \sin(2\pi x), & x < 0 \\ x^2 - cx, & x \geq 0 \end{cases}$$ ### Conditions for Continuity: For $$f(x)$$ to be continuous at $$x = 0$$, the following must hold: 1. $$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x)$$ 2. $$f(0)$$ must equal this common limit. ### Step 1: Evaluate $$\lim_{x \to 0^-} f(x)$$ For $$x < 0$$, we have: $$f(x) = e^x \sin(2\pi x)$$ As $$x \to 0^-$$, $$e^x \to 1$$ and $$\sin(2\pi x) \to 0$$. Therefore: $$\lim_{x \to 0^-} f(x) = 1 \times 0 = 0$$ ### Step 2: Evaluate $$\lim_{x \to 0^+} f(x)$$ For $$x \geq 0$$, we have: $$f(x) = x^2 - cx$$ At $$x = 0$$, the function becomes: $$f(0) = 0^2 - c \times 0 = 0$$ Thus, $$f(0) = 0$$. Now, we calculate the limit as $$x \to 0^+$$: $$\lim_{x \to 0^+} (x^2 - cx) = 0^2 - c \times 0 = 0$$ ### Step 3: Ensure Limits Match Since: $$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = 0$$ The function is already continuous for any value of $$c$$. ### Conclusion: The function $$f(x)$$ is continuous at $$x = 0$$ for **any value of $$c$$**. Thus, the correct option is: $$\boxed{\text{any value of } c}$$ --- Would you like further clarification or details? Here are some related questions for deeper understanding: 1. How do you verify continuity for piecewise functions? 2. What happens if the left-hand and right-hand limits do not match? 3. How would the continuity conditions change if the function had a jump discontinuity? 4. Can you give an example where a function is continuous but not differentiable at a point? 5. How do different trigonometric components affect limits near zero? **Tip:** Always check both left-hand and right-hand limits to ensure continuity at a point, especially for piecewise functions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Piecewise Functions
Formulas
lim_{x -> 0^-} f(x) = lim_{x -> 0^+} f(x)
f(x) = e^x * sin(2πx) for x < 0
f(x) = x^2 - cx for x >= 0
Theorems
Limit Definition of Continuity
Suitable Grade Level
Grade 11-12 (Pre-Calculus/Calculus)