\(\lim _{\mathrm{x} \rightarrow(\pi / 3)}[\\{\sin [(\mathrm{x} / 2)-(\pi / 6)]\\} /\\{2 \cos [(\mathrm{x} / 2)-(\pi / 2)]-1\\}]=?\) \(\begin{array}{llll}\text { (a) }-(1 / \sqrt{3}) & \text { (b) } \sqrt{3} & \text { (c) }(1 / \sqrt{3}) & \text { (d) }[(-1) /(2 \sqrt{3})]\end{array}\)

Understanding trigonometric identities is crucial when dealing with limits of trigonometric functions. Trigonometric identities are equations that hold true for any value of the variable within their domains. One basic identity is the Pythagorean identity, which states that for any angle \theta, \[\sin^2(\theta) + \cos^2(\theta) = 1.\] Another subset of identities are the reciprocity identities, such as \[\sec(\theta) = \frac{1}{\cos(\theta)},\] \[\csc(\theta) = \frac{1}{\sin(\theta)}.\] There are also quotient identities that involve tangent and cotangent, like \[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)},\] \[\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}.\]
These identities are pivotal because they allow us to simplify complex trigonometric expressions before finding limits. The ability to rewrite trigonometric functions using identities often makes it possible to perform direct substitution in limits, as seen in Step 2 of the provided solution where angle subtraction formulas are used to rewrite the sine and cosine terms.

Angle subtraction formulas are a specific type of trigonometric identity that express the sine or cosine of the difference between two angles in terms of the sines and cosines of the two angles. They are particularly useful when simplifying expressions involving trigonometric functions of sums or differences, making it easier to evaluate limits. For any angles \(a\) and \(b\), these formulas are given by:
\[\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b),\]
\[\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b).\]
In the problem at hand, these formulae transform the sine and cosine of angle differences into products of simpler trigonometric functions. As per Step 2 of the solution, using these formulas allowed us to expand the numerator and denominator into terms of \(\sin(\frac{x}{2})\) and \(\cos(\frac{x}{2})\), which we could then evaluate at the limit point. Without these angle subtraction formulas, determining the limit of the initial complex expression would have been significantly more challenging.

Direct substitution is often the most straightforward approach to evaluating limits of functions, including those involving trigonometric expressions. When a function is continuous at the point to which the variable is approaching, the limit can be found by simply plugging in the approaching value into the function.
To use direct substitution effectively, the function must not have any discontinuities such as holes, vertical asymptotes, or points of undefined value (like dividing by zero). If, upon substitution, we find the expression takes an indeterminate form such as \(0/0\) or \(\infty/\infty\), then other techniques such as L'Hôpital's rule or algebraic manipulation must be applied.
In Step 4 of the solution, direct substitution is employed after simplifying the trigonometric expression in the previous steps. It is essential first to ensure that the function is continuous at the point of interest. However, in the given exercise, the solution results in a value of 0, which does not match any option provided. This discrepancy suggests there may be a mistake in the provided exercise or answer choices. In such cases, revisiting the earlier steps and ensuring each trigonometric identity and formula was applied correctly is advisable.