Math Problem Statement
Tentukan koefisien c agar f(x) memiliki limit pada x = 4 dengan f(x) = {(3 - x)c, x < 4; x^2 + 2c, x ≥ 4}
Solution
The problem requires us to find the coefficient such that the function has a limit at . The function is defined as:
(3 - x)c, & x < 4 \\ x^2 + 2c, & x \geq 4 \end{cases}$$ To ensure the function has a limit at $$x = 4$$, we need the left-hand limit (as $$x \to 4^-$$) to be equal to the right-hand limit (as $$x \to 4^+$$) at $$x = 4$$. This requires that: $$\lim_{x \to 4^-} f(x) = \lim_{x \to 4^+} f(x)$$ ### Step 1: Left-hand Limit $$\lim_{x \to 4^-} f(x)$$ For $$x < 4$$, $$f(x) = (3 - x)c$$. $$\lim_{x \to 4^-} f(x) = \lim_{x \to 4^-} (3 - x)c = (3 - 4)c = -c$$ ### Step 2: Right-hand Limit $$\lim_{x \to 4^+} f(x)$$ For $$x \geq 4$$, $$f(x) = x^2 + 2c$$. $$\lim_{x \to 4^+} f(x) = \lim_{x \to 4^+} (x^2 + 2c) = 4^2 + 2c = 16 + 2c$$ ### Step 3: Set Left-hand Limit Equal to Right-hand Limit To have a limit at $$x = 4$$, we set the left-hand and right-hand limits equal to each other: $$-c = 16 + 2c$$ Solving for $$c$$: 1. Add $$c$$ to both sides: $$0 = 16 + 3c$$ 2. Subtract 16 from both sides: $$-16 = 3c$$ 3. Divide by 3: $$c = -\frac{16}{3}$$ ### Conclusion The value of $$c$$ that makes $$f(x)$$ have a limit at $$x = 4$$ is: $$c = -\frac{16}{3}$$ Would you like further details, or have any other questions? ### Related Questions 1. How do you determine the continuity of a piecewise function at a specific point? 2. What is the difference between a limit and continuity? 3. How does one find limits involving piecewise functions in general? 4. Why is it necessary for the left-hand limit to equal the right-hand limit for the existence of a limit? 5. How would the answer change if $$f(x)$$ were defined differently for $$x < 4$$ or $$x \geq 4$$? ### Tip In piecewise functions, always analyze each section individually when evaluating limits and continuity.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Formulas
Limit from the left (left-hand limit)
Limit from the right (right-hand limit)
Theorems
Limit existence theorem for piecewise functions
Suitable Grade Level
Grades 10-12