If \(\mathrm{f}(\mathrm{x})=[\\{\tan [(\pi / 6)-\mathrm{x}]\\} /(\operatorname{Cot} 3 \mathrm{x})] ; \mathrm{x} \neq(\pi / 6)\), is continuous at \(\mathrm{x}=(\pi / 6)\) then \(\mathrm{f}(\pi / 6)=\ldots \ldots\) (a) \([1 /(3 \sqrt{3})]\) (b) \((\sqrt{3} / 2)\) (c) \((1 / 3)\) (d) \([1 /(6 \sqrt{3})]\)

Limits and continuity are fundamental concepts in calculus that deal with the behavior of functions as they approach a certain point. For a function to be continuous at a point, it must meet three criteria: the function is defined at the point, the limit of the function as it approaches the point exists, and the limit is equal to the function's value at that point.

When evaluating limits, particularly when approaching a point where the function is not readily evaluated, mathematicians use various techniques to analyze the behavior. These techniques include algebraic manipulation, graphical analysis, and special limit rules such as factoring, conjugate multiplication, or limit laws. Understanding the concepts of limits and continuity helps students analyze the behavior of functions and find values of functions at points where direct substitution is not possible.

What is L'Hôpital's Rule?

L'Hôpital's Rule provides us with a powerful tool for evaluating limits that result in indeterminate forms, such as \(0/0\) or \(\infty/\infty\). This rule states if the limit \(\lim_{x \to c} f(x)/g(x)\) results in an indeterminate form, and both functions \(f(x)\) and \(g(x)\) are differentiable near \(c\), then the limit can be found by differentiating the numerator and denominator separately and taking the limit of their quotient.

It's important to remember that L'Hôpital's Rule can only be used under certain conditions - when the original limit results in an indeterminate form and when the derivatives of the numerator and the denominator are continuous around the point c. In practice, one may need to apply L'Hôpital's Rule more than once if the limit remains in an indeterminate form after the first application.

Trigonometric functions are periodic functions that are foundational to the study of triangles, waves, and many other applications in various fields. In calculus, the trigonometric functions—sine \(\sin\), cosine \(\cos\), tangent \(\tan\), cotangent \(\cot\), secant \(\sec\), and cosecant \(\csc\)—often appear in problems involving limits and continuity.

When working with trigonometric functions, it is important to understand their properties, such as periodicity, symmetry, and the values at key points on the unit circle. For example, \(\tan\) and \(\cot\) have a period of \(\pi\), while \(\sin\) and \(\cos\) have a period of \(2\pi\). Moreover, knowing the trigonometric identities, like pythagorean identities and angle sum and difference identities, can greatly simplify the process of evaluating limits and solve continuity problems.