https://file.aiquickdraw.com/mathbot/file/172991529315944ai0yu8.jpg

The problem asks to determine the coefficient \( c \) so that the function \( f(x) \) has a limit as \( x \) approaches 3.

The function \( f(x) \) is given as:
\[
f(x) = \begin{cases} 
ax + 2, & x < 3 \\
x^2 + c, & x \geq 3 
\end{cases}
\]

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.

Tentukan koefisien c agar f(x) memiliki limit di x = 3, dengan fungsi f(x) didefinisikan sebagai: 
\[
f(x) = \begin{cases} 
ax + 2, & x < 3 \\
x^2 + c, & x \geq 3 
\end{cases}
\]

Determine the coefficient c so that the function f(x) has a limit as x approaches 3. This involves evaluating left-hand and right-hand limits of a piecewise function and setting them equal for continuity. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation