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Understanding the concept of limits is crucial when dealing with functions that are not well-defined at certain points. A limit attempts to define the value that a function approaches as the input (or 'x' value) gets closer to a specific point.

For instance, consider the function \( g(x) = \frac{\sin x}{x} \). At \(x=0\), the function seems to be undefined because we cannot divide by zero. However, the concept of a limit allows us to explore what value the function approaches as \(x\) inches towards zero. To establish this, we often depict it as \( \lim_{x \to 0} g(x) \).

In this case, simply plugging in \(x=0\) is not possible. This is where L'Hospital's Rule comes into play. The rule applies when we encounter an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). By differentiating the numerator and denominator separately and then examining their limit, we can determine the original limit. Thus, as per L'Hospital's Rule, the limit for \( g(x) \) as \(x\) approaches zero is 1, transforming an undefined expression into a meaningful value.

The sine function is one of the basic trigonometric functions that describes a repetitive oscillation, like the movement of a pendulum or the waves of the sea. The function \( \sin x \) oscillates between -1 and 1, and is defined for all real numbers, making it continuous and smooth.

When examining the function \( g(x) = \frac{\sin x}{x} \), the sine component has well-known behavior. As \(x\) moves away from zero, the sine function's amplitude does not increase. Instead, the value of the sine function begins to cancel out more as \(x\) becomes larger because you're dividing by a larger number. This makes the function's graph approach zero as \(x\) heads towards infinity or negative infinity, highlighting the unique behavior of sine as part of the quotient in \( g(x) \).

At the point where \(x=0\), the sine function contributes to an interesting scenario where the 'undetermined' quotient of \( \frac{\sin x}{x} \) becomes 'determined' through the use of limits, and hence, L'Hospital's rule.

The continuity of trigonometric functions such as sine is a fundamental concept in understanding their behavior. A function is continuous at a point if the function's value at that point equals the limit of the function as it approaches that point.

Trigonometric functions, including the sine function, are continuous everywhere on their domain. This implies that there are no abrupt jumps, gaps, or breaks in their graphs. Even when dealing with the function \( g(x) = \frac{\sin x}{x} \)—despite the apparent complication when \(x\) is zero—continuity can be maintained by defining \( g(0) \) to be the limit of \( g(x) \) as \(x\) goes to zero.

This technique of 'filling in the hole' ensures that the entire function is continuous, including at the troublesome point where \(x\) is zero. Overall, this concept plays a pivotal role in calculus and forms the basis of understanding more complex behaviors in mathematical analysis.