91Ó°ÊÓ

L'Hôpital's Rule is a powerful tool in calculus used to find limits of indeterminate forms like \( 0/0 \) or \( \infty/\infty \). If faced with such a form, this rule allows you to differentiate the numerator and denominator separately and then take the limit of their quotient. Remember, it's essential that:

• The original limit results in an indeterminate form.
• Both functions are differentiable near the point of interest.
• The new limit of the derivatives exists.

For example, in the problem \( \lim_{x \to 0} \frac{1 - \cos 2x}{3x} \), directly substituting \( x = 0 \) results in \( 0/0 \), guiding us to use L'Hôpital's Rule. Differentiating the numerator and denominator gives us two new functions, \( 2\sin 2x \) for the numerator and \( 3 \) for the denominator. The new limit is then \( \lim_{x \to 0} \frac{2\sin 2x}{3} \), which simplifies to 0, giving us a clean solution.

Indeterminate forms are expressions encountered in calculus that do not straightforwardly suggest a limit. Common indeterminate forms include \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0 \times \infty \), \( \infty - \infty \), \( 0^0 \), \( 1^\infty \), and \( \infty^0 \). These require careful mathematical treatment.

• When encountering \( \frac{0}{0} \), it suggests that both the numerator and denominator approach zero. Direct evaluation won't yield a meaningful result.
• L'Hôpital's Rule, algebraic manipulation, or series expansion techniques can be employed to resolve these forms.

For our problem, \( \lim_{x \to 0} \frac{1 - \cos 2x}{3x} \) yielded a \( \frac{0}{0} \) form, specifically calling for L'Hôpital's Rule. The key is acknowledging when an expression doesn't directly provide a limit and using appropriate calculus tools to address it.

Trigonometric limits often appear in calculus because trigonometric functions like \( \sin x \) and \( \cos x \) have unique properties. When evaluating limits involving trigonometric functions, remember:

• \( \sin x \to 0 \) and \( \cos x \to 1 \) as \( x \to 0 \).
• Know special limits, such as \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \).
• Use identities (like \( \sin 2x = 2\sin x \cos x \)) to simplify problems where necessary.

In our exercise, the limit \( \lim_{x \to 0} \frac{2\sin 2x}{3} \) benefits from knowledge about \( \sin x \), as \( \sin 2x \to 0 \) mirrors \( \sin x \to 0 \). Recognizing the behavior of these functions as \( x \to 0 \) is crucial for evaluating such limits successfully.