The value of ordered pair (a,b) such that `lim _(xto0) (x (1+ a cos x ) -b sin x )/( x ^(3))=1,` is:

Options

A. `(-(5)/(2),-(3)/(2))`
B. `((5)/(2), (3)/(2))`
C. `(-(5)/(2 ),(3 )/(2))`
D. `((5 )/(2), -(3)/(2))`

Answer

To solve the limit problem, we need to find the ordered pair \((a, b)\) such that:

\[ \lim_{x \to 0} \frac{x(1 + a \cos x - b \sin x)}{x^3} = 1 \]

### Step-by-Step Solution:

1. **Rewrite the limit expression**:
We start with the limit expression:
\[ \lim_{x \to 0} \frac{x(1 + a \cos x - b \sin x)}{x^3} \]
This can be simplified to:
\[ \lim_{x \to 0} \frac{1 + a \cos x - b \sin x}{x^2} \]

2. **Expand \(\cos x\) and \(\sin x\)**:
Using Taylor series expansions around \(x = 0\):
\[ \cos x \approx 1 - \frac{x^2}{2} + O(x^4) \]
\[ \sin x \approx x - \frac{x^3}{6} + O(x^5) \]
Substituting these expansions into the limit gives:
\[ 1 + a\left(1 - \frac{x^2}{2}\right) - b\left(x - \frac{x^3}{6}\right) \]
This simplifies to:
\[ 1 + a - b x - \frac{a x^2}{2} + \frac{b x^3}{6} \]

3. **Combine terms**:
Now we can rewrite the expression:
\[ \lim_{x \to 0} \frac{(1 + a) - b x - \frac{a x^2}{2} + \frac{b x^3}{6}}{x^2} \]

4. **Separate the limit**:
As \(x \to 0\), the term \((1 + a)\) must equal \(0\) for the limit to exist (otherwise, the limit would approach infinity). Thus:
\[ 1 + a - b = 0 \quad \text{(Equation 1)} \]

5. **Evaluate the remaining terms**:
The remaining terms give:
\[ \lim_{x \to 0} \left(-\frac{a}{2} + \frac{b}{6} \right) = 1 \]
This leads to:
\[ -\frac{a}{2} + \frac{b}{6} = 1 \quad \text{(Equation 2)} \]

6. **Solve the equations**:
From Equation 1:
\[ a - b = -1 \implies b = a + 1 \]
Substitute \(b\) in Equation 2:
\[ -\frac{a}{2} + \frac{a + 1}{6} = 1 \]
Multiply through by 6 to eliminate the fractions:
\[ -3a + (a + 1) = 6 \]
Simplifying gives:
\[ -2a + 1 = 6 \implies -2a = 5 \implies a = -\frac{5}{2} \]
Now substituting \(a\) back to find \(b\):
\[ b = -\frac{5}{2} + 1 = -\frac{3}{2} \]

7. **Final ordered pair**:
Thus, the ordered pair \((a, b)\) is:
\[ \left(-\frac{5}{2}, -\frac{3}{2}\right) \]