Evaluate `lim_(xto0)(cosecx)^(x)`.

Answer

To evaluate the limit \( \lim_{x \to 0} (\csc x)^x \), we will follow these steps:

### Step 1: Set up the limit
Let \( y = (\csc x)^x \). We want to find \( \lim_{x \to 0} y \).

### Step 2: Take the natural logarithm
Taking the natural logarithm on both sides, we have:
\[ \ln y = x \ln (\csc x) \]

### Step 3: Evaluate the limit of the logarithm
Now we need to evaluate:
\[ \lim_{x \to 0} \ln y = \lim_{x \to 0} x \ln (\csc x) \]
As \( x \to 0 \), \( \csc x = \frac{1}{\sin x} \) approaches infinity, and thus \( \ln (\csc x) \) approaches infinity. Therefore, we have the indeterminate form \( 0 \cdot \infty \).

### Step 4: Rewrite the limit
To resolve the indeterminate form, we rewrite the limit:
\[ \lim_{x \to 0} x \ln (\csc x) = \lim_{x \to 0} \frac{\ln (\csc x)}{\frac{1}{x}} \]
This transforms our limit into the form \( \frac{\infty}{\infty} \).

### Step 5: Apply L'Hôpital's Rule
Now we can apply L'Hôpital's Rule:
\[ \lim_{x \to 0} \frac{\ln (\csc x)}{\frac{1}{x}} = \lim_{x \to 0} \frac{\frac{d}{dx} \ln (\csc x)}{\frac{d}{dx} \left(\frac{1}{x}\right)} \]
Calculating the derivatives:
- The derivative of \( \ln (\csc x) \) is \( -\cot x \).
- The derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \).

Thus, we have:
\[ \lim_{x \to 0} \frac{-\cot x}{-\frac{1}{x^2}} = \lim_{x \to 0} \frac{x^2 \cot x}{1} = \lim_{x \to 0} x^2 \frac{\cos x}{\sin x} \]

### Step 6: Simplify the limit
As \( x \to 0 \), \( \frac{\sin x}{x} \to 1 \), so:
\[ \lim_{x \to 0} x^2 \frac{\cos x}{\sin x} = \lim_{x \to 0} x^2 \cdot \frac{1}{1} = 0 \]

### Step 7: Conclude the limit for \( y \)
Since \( \lim_{x \to 0} \ln y = 0 \), we can exponentiate to find \( y \):
\[ y = e^{\lim_{x \to 0} \ln y} = e^0 = 1 \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0} (\csc x)^x = 1 \]

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