$\displaystyle \lim_{t\to0} \frac{\tan(6t)}{\sin(2t)}$
The limit of a trigonometric function is the value that a function approaches as its input approaches a certain value. It is a fundamental concept in calculus that helps determine the behavior of a function near a specific point.
The limit of a trigonometric function can be found by evaluating the function at the point in question and observing the behavior of the function as the input approaches that point. If the function approaches a specific value, then that value is the limit. If the function oscillates or becomes undefined, then the limit does not exist.
A one-sided limit is the value that a function approaches as its input approaches a certain value from only one side (either from the left or right). A two-sided limit is the value that a function approaches as its input approaches a certain value from both the left and right sides. In some cases, these two values may be different, indicating that the limit does not exist.
The limit of a trigonometric function does not exist when the function oscillates or becomes undefined as the input approaches a certain value. This can happen when the function has a vertical asymptote or when the input approaches a point where the function is undefined (such as when dividing by 0).
The concept of a limit and its applications in trigonometry are used in various fields such as physics, engineering, and economics. For example, in physics, the velocity of an object can be determined by finding the limit of its position function as time approaches a certain value. In engineering, the limit can be used to analyze the stability and behavior of structures. In economics, limits can be used to study the behavior of supply and demand functions.