Math Problem Statement
Tentukan koefisien a agar fungsi f(x) kontinu di x = 0. Fungsi didefinisikan sebagai: f(x) = {x^2 + 6a, x ≤ 0 ; (sin 6x)/(3x), x > 0}
Solution
The problem asks to determine the coefficient such that the function is continuous at . The function is defined as:
\begin{cases} x^2 + 6a, & x \leq 0 \\ \frac{\sin 6x}{3x}, & x > 0 \end{cases}$$ To ensure continuity at $$x = 0$$, the following condition must be satisfied: $$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0)$$ ### Step 1: Evaluate $$f(0)$$ From the piecewise definition, for $$x \leq 0$$, we substitute $$x = 0$$ into the expression: $$f(0) = 0^2 + 6a = 6a$$ ### Step 2: Evaluate $$\lim_{x \to 0^+} f(x)$$ For $$x > 0$$, the function is $$\frac{\sin 6x}{3x}$$. We need to find: $$\lim_{x \to 0^+} \frac{\sin 6x}{3x}$$ Using the standard limit $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$, we apply the same idea here: $$\lim_{x \to 0^+} \frac{\sin 6x}{3x} = \frac{6}{3} = 2$$ ### Step 3: Set up the continuity condition For the function to be continuous at $$x = 0$$, we need: $$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0)$$ This gives the condition: $$6a = 2$$ ### Step 4: Solve for $$a$$ $$a = \frac{2}{6} = \frac{1}{3}$$ Thus, the value of $$a$$ that makes $$f(x)$$ continuous at $$x = 0$$ is $$\frac{1}{3}$$. Would you like more details on any step, or do you have any further questions? ### 5 Related Questions: 1. What is the general definition of continuity at a point? 2. How can the limit of trigonometric functions help solve continuity problems? 3. What are the conditions for a piecewise function to be continuous? 4. How does the Squeeze Theorem apply to limit calculations like $$\lim_{x \to 0} \frac{\sin x}{x}$$? 5. Can a function be continuous but not differentiable? What are examples of such functions? ### Tip: When checking the continuity of piecewise functions, always ensure the limits from both sides of the point of interest match the function's value at that point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
f(0) = 6a
lim (x -> 0^-) f(x) = lim (x -> 0^+) f(x) = f(0)
lim (x -> 0) (sin kx)/kx = 1 (applied limit)
Theorems
Limit Theorem for Trigonometric Functions
Continuity Conditions
Suitable Grade Level
Grades 11-12
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