Evaluate the following limits :
`Lim_( x to pi//6) (sin (x - pi/6))/(sqrt(3)/2 - cos x )`

Answer

To evaluate the limit

\[ L = \lim_{x \to \frac{\pi}{6}} \frac{\sin(x - \frac{\pi}{6})}{\frac{\sqrt{3}}{2} - \cos x} \]

we notice that directly substituting \( x = \frac{\pi}{6} \) gives us an indeterminate form \( \frac{0}{0} \). Therefore, we need to manipulate the expression to resolve this.

### Step 1: Change of Variables
Let \( h = x - \frac{\pi}{6} \). Then, as \( x \to \frac{\pi}{6} \), \( h \to 0 \). Thus, we can rewrite \( x \) as \( h + \frac{\pi}{6} \).

### Step 2: Rewrite the Limit
Substituting \( x \) in terms of \( h \):

\[ L = \lim_{h \to 0} \frac{\sin(h)}{\frac{\sqrt{3}}{2} - \cos(h + \frac{\pi}{6})} \]

### Step 3: Simplify the Denominator
Using the cosine addition formula, we have:

\[ \cos(h + \frac{\pi}{6}) = \cos h \cos \frac{\pi}{6} - \sin h \sin \frac{\pi}{6} \]

Knowing that \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \) and \( \sin \frac{\pi}{6} = \frac{1}{2} \), we can substitute these values:

\[ \cos(h + \frac{\pi}{6}) = \cos h \cdot \frac{\sqrt{3}}{2} - \sin h \cdot \frac{1}{2} \]

Thus, the denominator becomes:

\[ \frac{\sqrt{3}}{2} - \left( \frac{\sqrt{3}}{2} \cos h - \frac{1}{2} \sin h \right) = \frac{\sqrt{3}}{2} (1 - \cos h) + \frac{1}{2} \sin h \]

### Step 4: Substitute Back into the Limit
Now we substitute this back into our limit:

\[ L = \lim_{h \to 0} \frac{\sin(h)}{\frac{\sqrt{3}}{2} (1 - \cos h) + \frac{1}{2} \sin h} \]

### Step 5: Use Small Angle Approximations
As \( h \to 0 \), we can use the small angle approximations:

\[ \sin(h) \approx h \quad \text{and} \quad 1 - \cos(h) \approx \frac{h^2}{2} \]

Substituting these approximations gives:

\[ L = \lim_{h \to 0} \frac{h}{\frac{\sqrt{3}}{2} \cdot \frac{h^2}{2} + \frac{1}{2} h} \]

### Step 6: Simplify the Expression
This simplifies to:

\[ L = \lim_{h \to 0} \frac{h}{\frac{\sqrt{3}}{4} h^2 + \frac{1}{2} h} = \lim_{h \to 0} \frac{1}{\frac{\sqrt{3}}{4} h + \frac{1}{2}} \]

### Step 7: Evaluate the Limit
As \( h \to 0 \):

\[ L = \frac{1}{\frac{1}{2}} = 2 \]

Thus, the final answer is:

\[ \boxed{2} \]