91Ó°ÊÓ

In calculus, an indeterminate form refers to certain limits that lead to ambiguous expressions, often hinting that regular algebraic approaches may not work. Some of the common indeterminate forms that you may encounter include expressions like \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0 \times \infty \), and many others. In our example, as \(x\) approaches \(0^+\), both \(\ln(\sin x)\) and \(\ln(\tan x)\) head towards \(-\infty\). This results in the indeterminate form \(\frac{-\infty}{-\infty}\). The presence of an indeterminate form like this usually indicates the potential to apply L'Hopital's Rule to find the limit. Such forms appear because the behavior near a specific point is not easily defined, hence regular substitution would fail.

Limits are a foundational concept in calculus and are used to describe the behavior of a function as it approaches a certain point. When calculating a limit like \(\lim _{x \rightarrow 0^{+}} \frac{\ln (\sin x)}{\ln (\tan x)}\), we examine the value that the function approaches as \(x\) gets closer and closer to 0 from the right side (positive side).

• In our problem, as \(x\) nears 0, both \(\sin x\) and \(\tan x\) approach 0, and consequently, \(\ln(\sin x)\) and \(\ln(\tan x)\) approach \(-\infty\).
• This makes the situation complex because both the numerator and the denominator tend to infinity in a negative sense, resulting in an expression that cannot be evaluated directly through substitution.
• Therefore, L'Hopital's Rule is ideal in this scenario, as it allows us to differentiate the numerator and denominator separately to compute the limit more effectively.

Derivatives provide information about the rate at which a function changes. They are instrumental in applying L'Hopital's Rule, which involves differentiating both the numerator and the denominator of a limit. In our exercise, differentiating \(\ln(\sin x)\) results in \(\frac{1}{\sin x} \cdot \cos x\) because it applies the chain rule, reflecting how the concentration of \(\sin x\) affects its natural logarithm. Similarly, the derivative of \(\ln(\tan x)\) is derived as \(\frac{1}{\tan x} \cdot \frac{1}{\cos^{2}x}\), which follows from differentiating \(\tan x\) and applying the chain rule once more.

• The chain rule is used in both derivations, reflecting how functions nested within other functions influence their rates of change.
• After these derivatives are found, they are used to form a new fraction representing the limit, \(\lim_ {x \rightarrow 0^{+}} \frac{\cos x}{\sin x} \cdot \frac{1}{\cos^{2}x}\), which can then be simplified to further investigate the limit as \(x \to 0^+\).