`lim_(xto1) [cosec(pix)/(2)]^(1//(1-x))` (where `[.]` represents the greatest integer function) is equal to

Options

A. (i) exists, (ii) does not exist
B. (i) does not exist, (ii) exists
C. both (i) and (ii) exist
D. neither (i) nor (ii) exists

Explanation

To solve the limit \( \lim_{x \to 1} \left[ \frac{\csc(\pi x)}{2} \right]^{\frac{1}{1-x}} \), where \([.]\) represents the greatest integer function, we can follow these steps:

### Step 1: Analyze the limit as \( x \to 1 \)

As \( x \) approaches 1, we need to evaluate \( \csc(\pi x) \):
\[ \csc(\pi x) = \frac{1}{\sin(\pi x)} \]
When \( x = 1 \), \( \sin(\pi x) = \sin(\pi) = 0 \). Therefore, \( \csc(\pi x) \) approaches infinity as \( x \) approaches 1.

### Step 2: Substitute the limit into the expression

Now, we substitute this into our limit:
\[ \lim_{x \to 1} \left[ \frac{\csc(\pi x)}{2} \right]^{\frac{1}{1-x}} = \lim_{x \to 1} \left[ \frac{1}{2 \sin(\pi x)} \right]^{\frac{1}{1-x}} \]

### Step 3: Determine the behavior of \( \sin(\pi x) \)

As \( x \to 1 \), \( \sin(\pi x) \) approaches 0. Therefore, \( \frac{1}{\sin(\pi x)} \) approaches infinity, which means \( \frac{\csc(\pi x)}{2} \) also approaches infinity.

### Step 4: Evaluate the limit of the exponent

Now, we need to analyze the exponent \( \frac{1}{1-x} \):
- As \( x \to 1 \), \( 1-x \) approaches 0, thus \( \frac{1}{1-x} \) approaches \( +\infty \).

### Step 5: Combine the results

Now we have an expression of the form \( \infty^{\infty} \):
\[ \lim_{x \to 1} \left[ \frac{\csc(\pi x)}{2} \right]^{\frac{1}{1-x}} = \infty^{\infty} \]

### Step 6: Apply the greatest integer function

Since \( \left[ \frac{\csc(\pi x)}{2} \right] \) approaches \( \infty \) as \( x \to 1 \), we can conclude that:
\[ \left[ \frac{\csc(\pi x)}{2} \right] \to \infty \]

### Step 7: Final limit evaluation

Thus, the limit becomes:
\[ \lim_{x \to 1} \left[ \frac{\csc(\pi x)}{2} \right]^{\frac{1}{1-x}} = \infty^{\infty} = 1 \]

### Conclusion

The final result is:
\[ \lim_{x \to 1} \left[ \frac{\csc(\pi x)}{2} \right]^{\frac{1}{1-x}} = 1 \]