\(\lim _{\mathrm{x} \rightarrow 1}\left[\left(\mathrm{x}^{365}-365 \mathrm{x}+364\right) /(\mathrm{x}-1)^{2}\right]=?\) (a) 66,430 (b) 64,340 (c) 66,630 (d) 64,430

When dealing with calculus, specifically limits, we often encounter expressions that lead to indeterminate forms. An indeterminate form is an undefined expression that does not have a unique limit or may have multiple limits. The most common indeterminate form is \( \frac{0}{0} \), as seen in the provided exercise. Others include \( \frac{\infty}{\infty} \), \( 0 \times \infty \), \( \infty - \infty \), \( 1^\infty \), and \( 0^0 \).

Understanding indeterminate forms is crucial since they inform us that more investigation is needed to find the limit of a function. It indicates that direct substitution will not work, and we must apply techniques like factoring, rationalization, or, as in the given problem, L'Hopital's Rule, to evaluate these limits properly.

A limit in calculus is essentially the value that a function approaches as the input approaches some point. Limits are foundational to the study of calculus because they underpin the definitions of continuity, derivatives, and integrals. Finding the limit of a function like \( \lim_{{x \rightarrow a}} f(x) \) often involves seeing what value f(x) will approach as x gets arbitrarily close to a.

However, when the function produces an indeterminate form, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), direct substitution isn't possible. That's where L'Hopital's Rule comes into play—it gives us a reliable method to evaluate many limits that at first seem uncertain.

Calculus is a significant part of the Mathematics syllabus for the Joint Entrance Examination (JEE), which is an engineering entrance assessment conducted in India. It's imperative for students preparing for the JEE to grasp concepts like differentiation and limits comprehensively, as they are the bedrock of advanced studies in physics and engineering.

L'Hopital's Rule, a topic often featured in JEE prep materials, is especially useful for addressing limit problems that result in indeterminate forms. Mastering this rule not only aids in solving textbook problems but also prepares students for the conceptual understanding required in higher education and practical applications in engineering.

In calculus, differentiation is the process of finding the derivative of a function, which represents the rate at which the function's value changes with respect to a change in its input value. Differentiation is a powerful tool used to solve a myriad of problems, ranging from simple motion to optimizing complex systems.

Within the context of L'Hopital's Rule, differentiation is used to simplify the evaluation of limits involving indeterminate forms. By taking the derivative of both the numerator and the denominator separately, we can convert an indeterminate form to a determinate one, which can then be more easily evaluated by direct substitution, as illustrated in the exercise's step-by-step solution.