Math Problem Statement
Tentukan koefisien c agar f(x) memiliki limit di x = 3, dengan fungsi f(x) didefinisikan sebagai:
\[
f(x) = \begin{cases}
ax + 2, & x < 3 \\
x^2 + c, & x \geq 3
\end{cases}
\]
Solution
The problem asks to determine the coefficient so that the function has a limit as approaches 3.
The function is given as:
ax + 2, & x < 3 \\ x^2 + c, & x \geq 3 \end{cases}$$ For $$f(x)$$ to have a limit at $$x = 3$$, the left-hand limit and the right-hand limit as $$x$$ approaches 3 must be equal. ### Step 1: Left-hand limit as $$x \to 3^-$$ Since $$x < 3$$, we use the expression $$ax + 2$$: $$\lim_{x \to 3^-} f(x) = a \cdot 3 + 2 = 3a + 2$$ ### Step 2: Right-hand limit as $$x \to 3^+$$ Since $$x \geq 3$$, we use the expression $$x^2 + c$$: $$\lim_{x \to 3^+} f(x) = 3^2 + c = 9 + c$$ ### Step 3: Equating the limits To ensure $$f(x)$$ has a limit at $$x = 3$$, set the left-hand limit equal to the right-hand limit: $$3a + 2 = 9 + c$$ ### Step 4: Solving for $$c$$ in terms of $$a$$ Rearrange the equation to solve for $$c$$: $$c = 3a - 7$$ So, the coefficient $$c$$ should be $$3a - 7$$ to make $$f(x)$$ continuous at $$x = 3$$ and ensure it has a limit there. Would you like further explanation or examples related to continuity and limits? --- 1. What are left-hand and right-hand limits in continuity problems? 2. How do you determine the continuity of a piecewise function? 3. Why do limits matter in determining continuity at a point? 4. Can you explain more about solving equations involving two variables? 5. How can we confirm if a limit exists in piecewise functions? **Tip:** For a piecewise function to have a limit at a point, ensure both expressions approach the same value at that point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Continuity
Piecewise Functions
Formulas
Limit of a piecewise function
Solving for continuity at a point
Theorems
Limit theorem for continuity
Suitable Grade Level
Grades 10-12