Math Problem Statement
Tentukan koefisien a agar fungsi f(x) kontinu di x = 0 dengan f(x) didefinisikan sebagai: f(x) = {x^2 + 6a, x ≤ 0; (sin(6x))/(3x), x > 0}.
Solution
The problem asks to determine the coefficient so that the function is continuous at . The function is piecewise-defined as:
x^2 + 6a, & x \leq 0 \\ \frac{\sin(6x)}{3x}, & x > 0 \end{cases}$$ For the function to be continuous at $$x = 0$$, the left-hand limit (as $$x$$ approaches 0 from the left) must equal the right-hand limit (as $$x$$ approaches 0 from the right), and both must equal the function value at $$x = 0$$. ### Step 1: Evaluate the left-hand limit For $$x \leq 0$$, the function is $$f(x) = x^2 + 6a$$. So, the limit as $$x \to 0^-$$ is: $$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (x^2 + 6a) = 6a$$ ### Step 2: Evaluate the right-hand limit For $$x > 0$$, the function is $$f(x) = \frac{\sin(6x)}{3x}$$. To find the limit as $$x \to 0^+$$, we use the fact that $$\lim_{x \to 0} \frac{\sin(kx)}{x} = k$$. Here, $$k = 6$$, so: $$\lim_{x \to 0^+} \frac{\sin(6x)}{3x} = \frac{6}{3} = 2$$ ### Step 3: Set the limits equal for continuity For the function to be continuous at $$x = 0$$, the left-hand and right-hand limits must be equal: $$6a = 2$$ ### Step 4: Solve for $$a$$ $$a = \frac{2}{6} = \frac{1}{3}$$ Thus, the coefficient $$a$$ must be $$\frac{1}{3}$$ for the function to be continuous at $$x = 0$$. Would you like more details or have any questions? Here are 5 related questions you might consider: 1. What is the general definition of continuity at a point? 2. How would the process change if the piecewise function had a different form for $$x > 0$$? 3. Can you provide an example of a function that is not continuous at a point? 4. What is the importance of limits in determining the continuity of a function? 5. How does the squeeze theorem apply to functions like $$\frac{\sin(kx)}{x}$$ as $$x \to 0$$? **Tip:** When solving continuity problems with trigonometric functions, remember the standard limits involving sine and cosine, such as $$\lim_{x \to 0} \frac{\sin(kx)}{x} = k$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Piecewise Functions
Formulas
Continuity at a point: f(x) is continuous at x = c if lim_{x→c^-} f(x) = lim_{x→c^+} f(x) = f(c).
Limit of sine function: lim_{x→0} (sin(kx))/x = k.
Theorems
Limit Theorem
Squeeze Theorem
Suitable Grade Level
Grades 11-12
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