Find the values of a and b if `lim_(xto2)f(x) and lim_(xto4)f(x)` exists where
`f(x)={: (x^(2)+ax+b", "0lexlt2),(3x+2", "2lexle4),(2ax+5b", "4ltxlt8):}`

Answer

To find the values of \( a \) and \( b \) such that the limits \( \lim_{x \to 2} f(x) \) and \( \lim_{x \to 4} f(x) \) exist, we need to ensure that the left-hand limit and right-hand limit at these points are equal.

### Step 1: Finding the limit as \( x \to 2 \)

For \( x \) approaching 2, we have:

- Left-hand limit (\( x \to 2^- \)):
\[ f(x) = x^2 + ax + b \]
Evaluating at \( x = 2 \):
\[ \lim_{x \to 2^-} f(x) = 2^2 + 2a + b = 4 + 2a + b \]

- Right-hand limit (\( x \to 2^+ \)):
\[ f(x) = 3x + 2 \]
Evaluating at \( x = 2 \):
\[ \lim_{x \to 2^+} f(x) = 3(2) + 2 = 6 + 2 = 8 \]

Setting the left-hand limit equal to the right-hand limit:
\[ 4 + 2a + b = 8 \]
This simplifies to:
\[ 2a + b = 4 \quad \text{(Equation 1)} \]

### Step 2: Finding the limit as \( x \to 4 \)

For \( x \) approaching 4, we have:

- Left-hand limit (\( x \to 4^- \)):
\[ f(x) = 3x + 2 \]
Evaluating at \( x = 4 \):
\[ \lim_{x \to 4^-} f(x) = 3(4) + 2 = 12 + 2 = 14 \]

- Right-hand limit (\( x \to 4^+ \)):
\[ f(x) = 2ax + 5b \]
Evaluating at \( x = 4 \):
\[ \lim_{x \to 4^+} f(x) = 2a(4) + 5b = 8a + 5b \]

Setting the left-hand limit equal to the right-hand limit:
\[ 8a + 5b = 14 \quad \text{(Equation 2)} \]

### Step 3: Solving the equations

Now we have a system of equations:
1. \( 2a + b = 4 \)
2. \( 8a + 5b = 14 \)

From Equation 1, we can express \( b \) in terms of \( a \):
\[ b = 4 - 2a \]

Substituting \( b \) into Equation 2:
\[ 8a + 5(4 - 2a) = 14 \]
Expanding this:
\[ 8a + 20 - 10a = 14 \]
Combining like terms:
\[ -2a + 20 = 14 \]
Subtracting 20 from both sides:
\[ -2a = -6 \]
Dividing by -2:
\[ a = 3 \]

Now substituting \( a = 3 \) back into Equation 1 to find \( b \):
\[ 2(3) + b = 4 \]
This simplifies to:
\[ 6 + b = 4 \]
Subtracting 6 from both sides:
\[ b = -2 \]

### Final Values
Thus, the values of \( a \) and \( b \) are:
\[ a = 3, \quad b = -2 \]