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We are given the following piecewise function \( f(x) \):

\[
f(x) = 
\begin{cases}
2x + 3a & \text{if } x < 1 \\
a + 4 & \text{if } x = 1 \\
\sqrt{bx} & \text{if } x > 1
\end{cases}
\]

We need to find values of \( a \) and \( b \) such that the function is continuous at \( x = 1 \).

### Step 1: Condition for Continuity
A function is continuous at \( x = 1 \) if:
\[
\lim_{x \to 1^-} f(x) = f(1) = \lim_{x \to 1^+} f(x)
\]

### Step 2: Left-hand limit \( \lim_{x \to 1^-} f(x) \)
For \( x < 1 \), \( f(x) = 2x + 3a \). Thus, we compute:
\[
\lim_{x \to 1^-} f(x) = 2(1) + 3a = 2 + 3a
\]

### Step 3: Value at \( x = 1 \)
From the function definition, \( f(1) = a + 4 \).

### Step 4: Right-hand limit \( \lim_{x \to 1^+} f(x) \)
For \( x > 1 \), \( f(x) = \sqrt{bx} \). Thus, we compute:
\[
\lim_{x \to 1^+} f(x) = \sqrt{b(1)} = \sqrt{b}
\]

### Step 5: Set up the Continuity Conditions
For the function to be continuous at \( x = 1 \), we need the following equalities:
1. \( \lim_{x \to 1^-} f(x) = f(1) \), i.e., \( 2 + 3a = a + 4 \)
2. \( f(1) = \lim_{x \to 1^+} f(x) \), i.e., \( a + 4 = \sqrt{b} \)

### Step 6: Solve the System of Equations
1. Solve \( 2 + 3a = a + 4 \):
   \[
   2 + 3a = a + 4 \implies 2a = 2 \implies a = 1
   \]
   
2. Substitute \( a = 1 \) into \( a + 4 = \sqrt{b} \):
   \[
   1 + 4 = \sqrt{b} \implies \sqrt{b} = 5 \implies b = 25
   \]

### Final Answer:
The values of \( a \) and \( b \) that make the function continuous at \( x = 1 \) are:
\[
a = 1, \quad b = 25
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How can you prove that a function is continuous at a different point, such as \( x = 2 \)?
2. What happens if the function was discontinuous at \( x = 1 \)? What kind of discontinuity would it be?
3. How do you find the derivative of a piecewise function like this?
4. How does the square root function \( \sqrt{bx} \) affect the overall continuity of the function?
5. How would the continuity condition change if we were dealing with limits approaching infinity?

**Tip:** When checking for continuity at a point, always ensure that both the left-hand and right-hand limits match the function value at that point.

Find the values of a and b that make the following function continuous at x = 1. f(x) = {2x + 3a if x < 1; a + 4 if x = 1; sqrt(bx) if x > 1}.

Learn how to find the values of a and b to make a piecewise function continuous at x = 1. This solution covers continuity, limits, and solving systems of equations for algebraic functions. Suitable for college-level calculus students.

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