91Ó°ÊÓ

L'Hôpital's Rule is a powerful tool in calculus for finding limits, especially when dealing with indeterminate forms like \(0/0\) or \( \infty/\infty \). It allows us to transform complex limit problems into simpler ones by differentiating the numerator and the denominator until a determinate form is achieved. This method is straightforward but requires that both the original function's numerator and denominator can be differentiated. The essence of L'Hôpital's Rule is:

• Identify the indeterminate form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
• Differentiates the numerator and the denominator separately.
• Recompute the limit of the new function resulting from the derivatives.
• If the new limit remains indeterminate, repeat the process.

In our exercise, we rearranged the sequence \( a_n = n(1 - \cos(1/n)) \) into the function \( f(x) = \frac{1 - \cos x}{x} \), when expressed in terms of \( x \). By applying L'Hôpital’s Rule, we differentiate and simplify our function to obtain a definitive limit of zero. This systematic approach makes it easier to tackle problems involving indeterminate forms.

Indeterminate forms occur when a limit is not immediately clear due to an undefined mathematical structure, like \(0/0\) or \(\infty/\infty\). These forms need special techniques to resolve, because conventional substitution in limits leads to ambiguity. Indeterminate forms signal that more work, like applying derivatives, is needed to find a limit.
In our sequence problem, when transformed to \(\frac{1-\cos x}{x}\), the limit \(\lim_{x \to 0} \) hits an indeterminate form \(\frac{0}{0}\).
This requires using L'Hôpital's Rule to differentiate the numerator and the denominator separately before recalculating the limit. Resolving such forms is crucial to simplifying expressions and verifying the existence of limits, as demonstrated in the process of reaching a limit of zero for our sequence.

The cosine function, a fundamental trigonometric function, is periodic and oscillates between -1 and 1. It includes important properties that influence limit calculation, such as:

• \( \cos(0) = 1 \)
• \(\cos(x)\) is even, satisfying \(\cos(-x) = \cos(x)\)

These properties are useful when dealing with limits and derivatives involving the cosine function.
In the context of our exercise, understanding cosine's behavior near zero is crucial. The small-angle approximation indicates \( 1 - \cos(x) \approx \frac{x^2}{2} \) for small values of \( x \), aiding in the simplification and differentiation needed to resolve indeterminate forms. This approximation, however, was not directly used here because L'Hôpital's Rule easily resolves the expression after determining the derivative of the cosine function \( \sin x \). Understanding these properties about cosine and how they behave in derivative form is vital when such functions are part of a limit problem.