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Understanding the behavior of trigonometric functions as variables approach a particular value is crucial in calculus. When evaluating the limits of trigonometric functions, several outcomes are possible.

For functions such as \( \sin x \) and \( \cos x \) as variables tend toward infinity, these functions do not have a limit because they oscillate indefinitely without approaching a single finite value. Specifically, the function \( \cos x \) has a range from -1 to 1 and it keeps oscillating within this range as \( x \) approaches infinity, as seen in the exercise part (a).

In contrast, when dealing with limits that approach a finite number, such as \( x \) approaching 0 from the positive direction in \( \frac{1}{\sin x} \), the limit can describe the function's behavior near the value. In this instance, the function tends towards positive infinity, which is captured in part (c) of the exercise.

In calculus, an infinite limit describes the behavior of a function as it increases or decreases without bound, approaching positive or negative infinity. Infinite limits often involve denominators approaching zero, as seen in functions like \( \frac{1}{x} \) where \( x \) approaches 0.

In part (c) of the given exercise, \( \frac{1}{\sin x} \) takes on increasingly large positive values as \( x \) approaches zero from the positive side, indicating an infinite limit. Remember, while we may say the limit is infinity, it's more accurate to state that the function 'diverges' or increases without bound, as infinity is not a number but rather a concept of unboundedness.

Trigonometric functions like \( \sin x \) and \( \cos x \) exhibit a repetitive pattern known as oscillation. The significance of this behavior is that as \( x \) goes to infinity or zero, these functions do not approach a particular value but continue to fluctuate within a specific range.

For example, as \( x \) approaches infinity in the function \( \cos x \) from part (a) of the exercise, there is no single value to which \( \cos x \) approaches and hence the limit does not exist. Similarly, \( \sin\left(\frac{1}{x}\right) \) in part (b) becomes increasingly wild as \( x \) approaches 0, causing rapid oscillations without approaching any particular value.

Dealing with limits that involve infinity often requires identifying the dominant terms in a function as the variable grows without bound. In expressions that involve polynomial functions or trigonometric functions this often means focusing on the highest degree of the variable in each term.

For instance, in part (e) of the exercise, inside the trigonometric function \( \cos \left(\frac{\pi x^{3}-99}{x^{3}-x^{2}+7}\right) \) the highest degree is the cubic term in both the numerator and the denominator. As \( x \) approaches infinity, the lesser degree terms become negligible and the leading coefficient of the cubic term dictates the limit of the function inside the cosine. Since the cosine function is continuous for all real numbers, the limit as \( x \) approaches infinity will be \( \cos \) of the constant to which the inside expression converges.