Math Problem Statement
Tentukan koefisien c agar f(x) memiliki limit di x = 3, dengan fungsi f(x) = {ax + 2, x < 3; x^2 + c, x ≥ 3}.
Solution
To solve this problem, we need to determine the value of that ensures the function has a limit at . The function is defined as:
ax + 2, & x < 3 \\ x^2 + c, & x \geq 3 \end{cases}$$ To ensure the limit exists at $$x = 3$$, the left-hand limit (as $$x$$ approaches 3 from the left) must equal the right-hand limit (as $$x$$ approaches 3 from the right), and both must equal $$f(3)$$. ### Step 1: Left-hand limit as $$x \to 3^-$$ For $$x < 3$$, we use the expression $$f(x) = ax + 2$$. $$\lim_{x \to 3^-} f(x) = a(3) + 2 = 3a + 2$$ ### Step 2: Right-hand limit as $$x \to 3^+$$ For $$x \geq 3$$, we use the expression $$f(x) = x^2 + c$$. $$\lim_{x \to 3^+} f(x) = 3^2 + c = 9 + c$$ ### Step 3: Equate the limits For $$f(x)$$ to have a limit at $$x = 3$$, the left-hand and right-hand limits must be equal: $$3a + 2 = 9 + c$$ ### Step 4: Solve for $$c$$ in terms of $$a$$ Rearrange the equation to solve for $$c$$: $$c = 3a - 7$$ Thus, the value of $$c$$ in terms of $$a$$ that ensures $$f(x)$$ has a limit at $$x = 3$$ is $$c = 3a - 7$$. --- Would you like further details on any part of this solution? Here are five related questions to explore: 1. What if $$a = 1$$? What value of $$c$$ would make the limit exist at $$x = 3$$? 2. How would we determine $$a$$ and $$c$$ if we wanted $$f(x)$$ to be continuous at $$x = 3$$? 3. What happens if we change the value of 2 in $$ax + 2$$ to another constant? How does this affect $$c$$? 4. What is the value of $$f(3)$$ in terms of $$c$$ and $$a$$? 5. How can we confirm continuity if $$a$$ and $$c$$ are already determined values? **Tip:** For a piecewise function to be continuous, ensure that the left and right limits at a boundary point are equal and match the function's value at that point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Continuity
Formulas
Limit of f(x) as x approaches a point
Solving for parameters to match limits
Theorems
Limit Existence at a Point
Suitable Grade Level
Grades 11-12