If `f(x)={{:(a+bcosx+csinx)/x^2,,xgt0), (9,,xge0):}}` is continuous at x = 0 , then the value of `(|a|+|b|)/5` is

Answer

To determine the value of \((|a| + |b|)/5\) for the function \(f(x) = \frac{a + b \cos x + c \sin x}{x^2}\) for \(x > 0\) and \(f(x) = 9\) for \(x = 0\), we need to ensure that the function is continuous at \(x = 0\). This means that the left-hand limit as \(x\) approaches 0 must equal the right-hand limit and the value of the function at that point.

### Step-by-Step Solution:

1. **Set up the limits for continuity**:
\[ \lim_{x \to 0^+} f(x) = f(0) \]
We know that \(f(0) = 9\).

2. **Calculate the right-hand limit**:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0} \frac{a + b \cos x + c \sin x}{x^2} \]
As \(x\) approaches 0, \(\cos x \to 1\) and \(\sin x \to 0\). Therefore, the limit becomes:
\[ \lim_{x \to 0} \frac{a + b \cdot 1 + c \cdot 0}{x^2} = \frac{a + b}{0} \]
This indicates that we have a \(0/0\) indeterminate form, which implies that \(a + b = 0\).

3. **Differentiate using L'Hôpital's Rule**:
Since we have a \(0/0\) form, we can apply L'Hôpital's Rule:
\[ \lim_{x \to 0} \frac{a + b \cos x + c \sin x}{x^2} = \lim_{x \to 0} \frac{-b \sin x + c \cos x}{2x} \]
Evaluating this limit as \(x\) approaches 0 gives:
\[ \frac{-b \cdot 0 + c \cdot 1}{0} = \frac{c}{0} \]
This is again an indeterminate form, so we need \(c = 0\) to avoid undefined behavior.

4. **Substituting \(c = 0\)**:
Now, we substitute \(c = 0\) back into our limit:
\[ \lim_{x \to 0} \frac{a + b \cos x}{x^2} = \lim_{x \to 0} \frac{a + b}{x^2} \]
Since \(a + b = 0\), we have:
\[ \lim_{x \to 0} \frac{0}{x^2} = 0 \]
Therefore, we need to apply L'Hôpital's Rule again:
\[ \lim_{x \to 0} \frac{-b \sin x}{2x} = \lim_{x \to 0} \frac{-b \cdot 0}{2} = 0 \]

5. **Setting the limit equal to the function value**:
We now set the limit equal to the function value at \(x = 0\):
\[ 0 = 9 \]
This is incorrect, indicating that we need to ensure that the limit approaches 9. Thus, we have:
\[ \frac{a}{2} = 9 \implies a = 18 \]

6. **Finding \(b\)**:
Since we established that \(a + b = 0\):
\[ 18 + b = 0 \implies b = -18 \]

7. **Calculate \((|a| + |b|)/5\)**:
\[ |a| = |18| = 18, \quad |b| = |-18| = 18 \]
Therefore,
\[ \frac{|a| + |b|}{5} = \frac{18 + 18}{5} = \frac{36}{5} = 7.2 \]

### Final Answer:
\[ \frac{|a| + |b|}{5} = 7.2 \]