`lim_(xrarr0^+)((cos^-1(x-[x]^2))*sin^-1((x-[x]^2)))/(x-x^3)`

Answer

To solve the limit

\[ \lim_{x \to 0^+} \frac{\cos^{-1}(x - [x]^2) \cdot \sin^{-1}(x - [x]^2)}{x - x^3} \]

we will break it down step by step.

### Step 1: Analyze the expression inside the limit

As \( x \to 0^+ \), we need to evaluate \( [x] \), which is the greatest integer function. For \( 0 < x < 1 \), we have \( [x] = 0 \). Therefore, \( [x]^2 = 0^2 = 0 \).

So, we can simplify \( x - [x]^2 \) to:

\[ x - [x]^2 = x - 0 = x \]

### Step 2: Substitute into the limit

Now, substituting this back into the limit, we have:

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

### Step 3: Simplify the denominator

Next, we simplify the denominator \( x - x^3 \):

\[ x - x^3 = x(1 - x^2) \]

### Step 4: Rewrite the limit

Now, our limit becomes:

\[ \lim_{x \to 0^+} \frac{\cos^{-1}(x) \cdot \sin^{-1}(x)}{x(1 - x^2)} \]

### Step 5: Evaluate the limit as \( x \to 0^+ \)

As \( x \to 0^+ \):

- \( \cos^{-1}(x) \to \cos^{-1}(0) = \frac{\pi}{2} \)
- \( \sin^{-1}(x) \to \sin^{-1}(0) = 0 \)

Thus, the product \( \cos^{-1}(x) \cdot \sin^{-1}(x) \) approaches:

\[ \frac{\pi}{2} \cdot 0 = 0 \]

### Step 6: Apply L'Hôpital's Rule

Since we have a \( \frac{0}{0} \) form, we can apply L'Hôpital's Rule. We differentiate the numerator and denominator:

1. **Numerator**:
- The derivative of \( \cos^{-1}(x) \) is \( -\frac{1}{\sqrt{1-x^2}} \)
- The derivative of \( \sin^{-1}(x) \) is \( \frac{1}{\sqrt{1-x^2}} \)

Using the product rule:

\[ \frac{d}{dx}[\cos^{-1}(x) \cdot \sin^{-1}(x)] = \cos^{-1}(x) \cdot \frac{1}{\sqrt{1-x^2}} + \sin^{-1}(x) \cdot \left(-\frac{1}{\sqrt{1-x^2}}\right) \]

2. **Denominator**:
- The derivative of \( x(1 - x^2) \) is \( 1 - 3x^2 \)

Now we can rewrite the limit:

\[ \lim_{x \to 0^+} \frac{\cos^{-1}(x) \cdot \frac{1}{\sqrt{1-x^2}} - \sin^{-1}(x) \cdot \frac{1}{\sqrt{1-x^2}}}{1 - 3x^2} \]

### Step 7: Evaluate the new limit

Now substituting \( x = 0 \):

- The numerator approaches \( \frac{\pi}{2} \cdot 1 - 0 \cdot 1 = \frac{\pi}{2} \)
- The denominator approaches \( 1 \)

Thus, the limit is:

\[ \frac{\frac{\pi}{2}}{1} = \frac{\pi}{2} \]

### Final Answer:

The limit is

\[ \frac{\pi}{2} \]