lim (sin x / sin a) cot(x - a) as x approaches a, where a is in (0, π/2)

Steps to Solve

1. Substitute the limit value

   Substitute $x = a$ into the expression: $$ \lim_{x \to a} \left(\frac{\sin x}{\sin a}\right) \cot(x - a) = \frac{\sin a}{\sin a} \cot(0) = 1 \cdot \cot(0) $$

2. Identify the limit form

   Since $\cot(0)$ is undefined (it approaches infinity), we must find another way to evaluate the limit using L'Hôpital's Rule or trigonometric identities.

3. Rewrite cotangent using sine and cosine

   The cotangent function can be rewritten: $$ \cot(x - a) = \frac{\cos(x - a)}{\sin(x - a)} $$

   Thus, the expression becomes: $$ \lim_{x \to a} \left(\frac{\sin x}{\sin a} \cdot \frac{\cos(x - a)}{\sin(x - a)}\right) $$

4. Use L'Hôpital's Rule

   Since both the numerator and denominator approach 0 as $x \to a$, we apply L'Hôpital's Rule: $$ \lim_{x \to a} \frac{\sin x \cdot \cos(x-a)}{\sin a \cdot \sin(x-a)} $$

5. Differentiate numerator and denominator

   Differentiate the numerator and denominator:

   • For the numerator: Derivative of $\sin x \cdot \cos(x - a)$
   • For the denominator: Derivative of $\sin a \cdot \sin(x - a)$

6. Evaluate the new limit

   Recalculate the limit: $$ \lim_{x \to a} \frac{\cos x \cdot \cos(x - a) - \sin x \cdot \sin(x - a)}{\sin a \cdot \cos(x - a)} $$

7. Final evaluation as $x \to a$

   Substitute $x = a$ into the newly derived limit expression to arrive at the solution.