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 solve the problem step by step, we need to ensure that the function \( f(x) \) is continuous at \( x = 0 \). The function is defined as:

\[ f(x) = \begin{cases} \frac{a + b \cos x + c \sin x}{x^2}, & x > 0 \\ 9, & x = 0 \end{cases} \]

### Step 1: Set up the continuity condition
For \( f(x) \) to be continuous at \( x = 0 \), we need:

\[ \lim_{x \to 0^+} f(x) = f(0) \]

Thus, we have:

\[ \lim_{x \to 0^+} \frac{a + b \cos x + c \sin x}{x^2} = 9 \]

### Step 2: Evaluate the limit
As \( x \to 0 \), \( \cos x \to 1 \) and \( \sin x \to 0 \). Therefore, we can substitute these values into the limit:

\[ \lim_{x \to 0^+} \frac{a + b \cdot 1 + c \cdot 0}{x^2} = \lim_{x \to 0^+} \frac{a + b}{x^2} \]

This expression becomes undefined (infinite) unless \( a + b = 0 \). Therefore, we must have:

\[ a + b = 0 \quad \text{(Equation 1)} \]

### Step 3: Substitute \( b \) in terms of \( a \)
From Equation 1, we can express \( b \) as:

\[ b = -a \]

### Step 4: Substitute back into the limit
Substituting \( b = -a \) into the limit gives:

\[ \lim_{x \to 0^+} \frac{a - a + c \sin x}{x^2} = \lim_{x \to 0^+} \frac{c \sin x}{x^2} \]

### Step 5: Apply L'Hôpital's Rule
Since this limit is also in the form \( \frac{0}{0} \), we can apply L'Hôpital's Rule:

\[ \lim_{x \to 0^+} \frac{c \sin x}{x^2} = \lim_{x \to 0^+} \frac{c \cos x}{2x} \]

### Step 6: Evaluate the new limit
As \( x \to 0 \), \( \cos x \to 1 \):

\[ \lim_{x \to 0^+} \frac{c \cdot 1}{2x} = \frac{c}{0} \]

This limit is undefined unless \( c = 0 \). Therefore, we conclude:

\[ c = 0 \quad \text{(Equation 2)} \]

### Step 7: Substitute \( c \) back into the limit
Now substituting \( c = 0 \) back into our limit expression gives:

\[ \lim_{x \to 0^+} \frac{a + b \cos x}{x^2} = \lim_{x \to 0^+} \frac{a - a \cdot 1}{x^2} = \lim_{x \to 0^+} \frac{0}{x^2} = 0 \]

### Step 8: Set the limit equal to \( f(0) \)
Now we know that:

\[ \lim_{x \to 0^+} f(x) = 9 \]

Thus, we have:

\[ \frac{a}{2} = 9 \]

### Step 9: Solve for \( a \)
From this, we find:

\[ a = 18 \]

### Step 10: Find \( b \)
Using \( b = -a \):

\[ b = -18 \]

### Step 11: Calculate \( \frac{|a| + |b|}{5} \)
Now we compute:

\[ \frac{|a| + |b|}{5} = \frac{|18| + |-18|}{5} = \frac{18 + 18}{5} = \frac{36}{5} = 7.2 \]

### Final Answer
Thus, the value of \( \frac{|a| + |b|}{5} \) is:

\[ \boxed{7.2} \]