\(\lim _{\mathrm{x} \rightarrow-(\pi / 4)}[\\{\sin \mathrm{x} \cdot \cos (5 \pi / 4)-\cos (7 \pi / 4) \cos \mathrm{x}\\} /(\pi+4 \mathrm{x})]\) \(=\) ? \(\begin{array}{lll}\text { (a) }-(1 / 3) & \text { (b) }(35 / 4) & \text { (c) }-(1 / 4)\end{array}\) (d) \(-(1 / 35)\)

The evaluation of trigonometric limits involves calculating the value that trigonometric functions approach as the input gets closer to a certain point. For students, understanding how to manipulate and simplify trigonometric expressions like \(\frac{\text{sin} x \text{cos}(5\text{pi}/4)-\text{cos}(7\text{pi}/4) \text{cos} x}{\text{pi}+4x}\) as seen in our exercise is crucial.

One key to mastering these limits is recognizing the standard trigonometric values, along with their properties. For instance, knowing that \(\text{cos}(5\text{pi}/4)\) and \(\text{cos}(7\text{pi}/4)\) are standard angles on the unit circle with well-known values simplifies the initial step.

Moreover, when analyzing trigonometric limits, it is critical to factor out common terms where possible, reducing the expression to a more manageable form. As demonstrated in the provided solution steps, factoring out \(-\frac{\text{sqrt}{2}}{2}\) results in a neater equation that can be easily handled in the limit evaluation process.

The limit of a function describes the behavior of the function as the input approaches a particular value. In the case of our example, evaluating the limit involves looking at what happens to the function \(\frac{\text{sin} x \text{cos}(5\text{pi}/4)-\text{cos}(7\text{pi}/4) \text{cos} x}{\text{pi}+4x}\) as \(x\) approaches \(-\frac{\text{pi}}{4}\).

Understanding this concept is paramount, as limits underpin many areas of calculus, including the derivation of derivatives and the analysis of function behaviors. When evaluating limits, continuity plays a significant role, as continuous functions allow for direct substitution – if a function is continuous at a point, the limit as the input approaches that point is the same as the function's value at that point. In the provided steps, because the trigonometric functions involved are continuous at the point of interest, we can confidently substitute \(-\frac{\text{pi}}{4}\) into the function to find the limit.

Continuous functions are functions that are uninterrupted or smooth; there are no jumps, breaks, or holes in their graphs. Such functions can be approached intuitively: if you can draw the function's graph without lifting your pencil from the paper, the function is continuous. This property is incredibly useful when calculating limits, as it ensures the behavior around the point of interest can be predicted by the function's value at that point.

The continuous nature of trigonometric functions such as \(\text{sin} x\) and \(\text{cos} x\) allows us to apply the limit laws, which include the direct substitution law, in our exercise. Here, plugging in \(-\frac{\text{pi}}{4}\) gives reliable results because we're assured there's no abrupt change in the function's value.

Remember, a solid grasp of continuity helps streamline many processes in calculus, as with our exercise, guiding students to the proper evaluation of limits and greater understanding of the function's behavior.