If `A=lim_(x to 0) (sin^(-1)(sinx))/(cos^(-1)(cosx))and B=lim_(x to 0)([|x|])/(x),` then

Options

A. A = 1
B. A does not exist
C. B = 0
D. B = 1

Explanation

To solve the given problem, we need to find the limits \( A \) and \( B \) as defined in the question. Let's break it down step by step.

### Step 1: Calculate \( A \)

We need to evaluate:
\[ A = \lim_{x \to 0} \frac{\sin^{-1}(\sin x)}{\cos^{-1}(\cos x)} \]

#### Step 1.1: Evaluate the left-hand limit (LHL)

For \( x \to 0^- \):
- When \( x < 0 \), \( \sin^{-1}(\sin x) = x \) (since \( \sin x \) is negative and \( \sin^{-1} \) will return the angle in the range \([- \frac{\pi}{2}, \frac{\pi}{2}]\)).
- For \( \cos^{-1}(\cos x) \), since \( x < 0 \), \( \cos^{-1}(\cos x) = -x \).

Thus, we have:
\[ \text{LHL} = \lim_{x \to 0^-} \frac{x}{-x} = -1 \]

#### Step 1.2: Evaluate the right-hand limit (RHL)

For \( x \to 0^+ \):
- When \( x > 0 \), \( \sin^{-1}(\sin x) = x \) (since \( \sin x \) is positive).
- For \( \cos^{-1}(\cos x) \), since \( x > 0 \), \( \cos^{-1}(\cos x) = x \).

Thus, we have:
\[ \text{RHL} = \lim_{x \to 0^+} \frac{x}{x} = 1 \]

#### Step 1.3: Compare LHL and RHL

Since:
\[ \text{LHL} = -1 \quad \text{and} \quad \text{RHL} = 1 \]
The limits do not match, hence:
\[ A \text{ does not exist.} \]

### Step 2: Calculate \( B \)

We need to evaluate:
\[ B = \lim_{x \to 0} \frac{|\lfloor x \rfloor|}{x} \]

#### Step 2.1: Evaluate the left-hand limit (LHL)

For \( x \to 0^- \):
- When \( x < 0 \), \( \lfloor x \rfloor = -1 \) (the greatest integer less than \( x \)).
- Hence, \( |\lfloor x \rfloor| = 1 \).

Thus, we have:
\[ \text{LHL} = \lim_{x \to 0^-} \frac{1}{x} \to -\infty \]

#### Step 2.2: Evaluate the right-hand limit (RHL)

For \( x \to 0^+ \):
- When \( x > 0 \), \( \lfloor x \rfloor = 0 \).
- Hence, \( |\lfloor x \rfloor| = 0 \).

Thus, we have:
\[ \text{RHL} = \lim_{x \to 0^+} \frac{0}{x} = 0 \]

#### Step 2.3: Compare LHL and RHL

Since:
\[ \text{LHL} = -\infty \quad \text{and} \quad \text{RHL} = 0 \]
The limits do not match, hence:
\[ B \text{ does not exist.} \]

### Final Results

- \( A \) does not exist.
- \( B \) does not exist.