Evaluate the limits :
`Lim_(theta to pi/6) (cot^(2) theta - 3)/(" cosec " theta - 2)`

Answer

To evaluate the limit

\[ \lim_{\theta \to \frac{\pi}{6}} \frac{\cot^2 \theta - 3}{\csc \theta - 2} \]

we will follow these steps:

### Step 1: Rewrite cotangent and cosecant in terms of sine and cosine
The cotangent and cosecant functions can be rewritten as follows:

\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \quad \text{and} \quad \csc \theta = \frac{1}{\sin \theta} \]

Thus,

\[ \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \quad \text{and} \quad \csc \theta = \frac{1}{\sin \theta} \]

### Step 2: Substitute the values into the limit expression
Substituting these into the limit gives:

\[ \lim_{\theta \to \frac{\pi}{6}} \frac{\frac{\cos^2 \theta}{\sin^2 \theta} - 3}{\frac{1}{\sin \theta} - 2} \]

### Step 3: Evaluate the limit directly
Now, we can evaluate the trigonometric functions at \(\theta = \frac{\pi}{6}\):

\[ \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \quad \text{and} \quad \sin \frac{\pi}{6} = \frac{1}{2} \]

Substituting these values gives:

\[ \cot^2 \frac{\pi}{6} = \left(\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}\right)^2 = \left(\sqrt{3}\right)^2 = 3 \]

Thus,

\[ \cot^2 \frac{\pi}{6} - 3 = 3 - 3 = 0 \]

And for cosecant:

\[ \csc \frac{\pi}{6} = \frac{1}{\frac{1}{2}} = 2 \]

So,

\[ \csc \frac{\pi}{6} - 2 = 2 - 2 = 0 \]

### Step 4: Apply L'Hôpital's Rule
Since both the numerator and denominator approach \(0\), we can apply L'Hôpital's Rule:

\[ \lim_{\theta \to \frac{\pi}{6}} \frac{\frac{d}{d\theta}(\cot^2 \theta - 3)}{\frac{d}{d\theta}(\csc \theta - 2)} \]

### Step 5: Differentiate the numerator and denominator
The derivative of the numerator:

\[ \frac{d}{d\theta}(\cot^2 \theta) = 2 \cot \theta (-\csc^2 \theta) = -2 \cot \theta \csc^2 \theta \]

The derivative of the denominator:

\[ \frac{d}{d\theta}(\csc \theta) = -\csc \theta \cot \theta \]

### Step 6: Substitute back into the limit
Now substituting back into the limit:

\[ \lim_{\theta \to \frac{\pi}{6}} \frac{-2 \cot \theta \csc^2 \theta}{-\csc \theta \cot \theta} \]

This simplifies to:

\[ \lim_{\theta \to \frac{\pi}{6}} 2 \csc \theta \]

### Step 7: Evaluate the limit
Now substituting \(\theta = \frac{\pi}{6}\):

\[ \csc \frac{\pi}{6} = 2 \]

Thus,

\[ 2 \cdot 2 = 4 \]

### Final Answer
Therefore, the limit is

\[ \boxed{4} \]