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

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Understand the Problem

The question is asking for the limit of a specific expression as x approaches a. It involves the sine function and cotangent function, with specified conditions for a. This is a calculus question related to limits.

Answer

The limit is $1$.
Answer for screen readers

The limit is $$ 1 $$.

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.

The limit is $$ 1 $$.

More Information

This limit relates to the behavior of sine and cotangent functions as they approach specific angles. Understanding how to manipulate trigonometric identities and the application of L'Hôpital's Rule in indeterminate forms is essential in calculus.

Tips

  • Forgetting to check for indeterminate forms before applying L'Hôpital's Rule.
  • Misapplying the derivative during differentiation of combined functions.
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