Let `L= lim_(x to 0)(a- sqrt(a^(2)-x^(2))-x^(2)/4)/x^(4), a gt 0`. If L is finite, then

Options

A. a=2
B. a=1
C. `L = 1/64`
D. `L =1/32`

Answer

To solve the limit problem \( L = \lim_{x \to 0} \frac{a - \sqrt{a^2 - x^2} - \frac{x^2}{4}}{x^4} \) where \( a > 0 \) and determine the conditions under which \( L \) is finite, we will follow these steps:

### Step 1: Identify the limit form
As \( x \to 0 \), both the numerator and denominator approach 0, resulting in the indeterminate form \( \frac{0}{0} \). This indicates that we can apply L'Hôpital's Rule.

### Step 2: Apply L'Hôpital's Rule
Using L'Hôpital's Rule, we differentiate the numerator and denominator separately.

1. **Differentiate the numerator**:
\[ \text{Numerator: } a - \sqrt{a^2 - x^2} - \frac{x^2}{4} \]
The derivative is:
\[ 0 - \frac{1}{2\sqrt{a^2 - x^2}}(-2x) - \frac{1}{2}x = \frac{x}{\sqrt{a^2 - x^2}} - \frac{x}{2} \]
Thus, the derivative of the numerator is:
\[ \frac{x}{\sqrt{a^2 - x^2}} - \frac{x}{2} \]

2. **Differentiate the denominator**:
\[ \text{Denominator: } x^4 \implies 4x^3 \]

### Step 3: Rewrite the limit
Now we can rewrite the limit using the derivatives:
\[ L = \lim_{x \to 0} \frac{\frac{x}{\sqrt{a^2 - x^2}} - \frac{x}{2}}{4x^3} \]

### Step 4: Simplify the expression
Factor out \( x \) from the numerator:
\[ L = \lim_{x \to 0} \frac{x\left(\frac{1}{\sqrt{a^2 - x^2}} - \frac{1}{2}\right)}{4x^3} = \lim_{x \to 0} \frac{\frac{1}{\sqrt{a^2 - x^2}} - \frac{1}{2}}{4x^2} \]

### Step 5: Apply L'Hôpital's Rule again
As \( x \to 0 \), we still have the form \( \frac{0}{0} \). We apply L'Hôpital's Rule again.

1. **Differentiate the new numerator**:
\[ \text{Numerator: } \frac{1}{\sqrt{a^2 - x^2}} - \frac{1}{2} \]
The derivative is:
\[ -\frac{1}{2}(a^2 - x^2)^{-3/2}(-2x) = \frac{x}{(a^2 - x^2)^{3/2}} \]

2. **Differentiate the denominator**:
\[ \text{Denominator: } 4x^2 \implies 8x \]

### Step 6: Rewrite the limit again
Now we have:
\[ L = \lim_{x \to 0} \frac{\frac{x}{(a^2 - x^2)^{3/2}}}{8x} = \lim_{x \to 0} \frac{1}{8(a^2 - x^2)^{3/2}} \]

### Step 7: Evaluate the limit
Substituting \( x = 0 \):
\[ L = \frac{1}{8(a^2)^{3/2}} = \frac{1}{8a^3} \]

### Step 8: Determine conditions for finiteness
For \( L \) to be finite, \( a \) must be greater than 0, which is already given in the problem.

### Conclusion
Thus, the limit \( L \) is finite if \( a > 0 \), and the value of \( L \) is:
\[ L = \frac{1}{8a^3} \]