Evaluate, `lim_(xto(pi//6)) (cot^(2)x-3)/("cosec"x-2)`

Options

A. `2`
B. `3`
C. `4`
D. `5`

Explanation

To evaluate the limit

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

we will follow these steps:

### Step 1: Substitute \( x = \frac{\pi}{6} \)

First, we need to check if substituting \( x = \frac{\pi}{6} \) gives us an indeterminate form.

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

Thus,

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

Now substitute into the limit:

\[ \cot^2\left(\frac{\pi}{6}\right) - 3 = 3 - 3 = 0, \]

and

\[ \csc\left(\frac{\pi}{6}\right) = \frac{1}{\sin\left(\frac{\pi}{6}\right)} = \frac{1}{\frac{1}{2}} = 2. \]

Thus,

\[ \csc\left(\frac{\pi}{6}\right) - 2 = 2 - 2 = 0. \]

So we have a \( \frac{0}{0} \) indeterminate form.

### Step 2: Use Trigonometric Identity

To resolve the indeterminate form, we can use the identity:

\[ \cot^2 x = \csc^2 x - 1. \]

Substituting this into our limit gives:

\[ \cot^2 x - 3 = \csc^2 x - 1 - 3 = \csc^2 x - 4. \]

Thus, we rewrite the limit as:

\[ \lim_{x \to \frac{\pi}{6}} \frac{\csc^2 x - 4}{\csc x - 2}. \]

### Step 3: Factor the Numerator

Notice that \( \csc^2 x - 4 \) can be factored as a difference of squares:

\[ \csc^2 x - 4 = (\csc x - 2)(\csc x + 2). \]

So we can rewrite the limit as:

\[ \lim_{x \to \frac{\pi}{6}} \frac{(\csc x - 2)(\csc x + 2)}{\csc x - 2}. \]

### Step 4: Cancel Common Factors

Now we can cancel \( \csc x - 2 \) from the numerator and the denominator (as long as \( x \neq \frac{\pi}{6} \)):

\[ \lim_{x \to \frac{\pi}{6}} (\csc x + 2). \]

### Step 5: Substitute Again

Now we can substitute \( x = \frac{\pi}{6} \):

\[ \csc\left(\frac{\pi}{6}\right) + 2 = 2 + 2 = 4. \]

### Final Answer

Thus, the limit evaluates to:

\[ \boxed{4}. \]