Exploring the Limit of Logarithmic Functions: Understanding the Value of lim_{x->0} log_{sinx} x

Mathematics often presents intriguing challenges when it comes to understanding the behavior of various functions as they approach certain values. One such challenge is exploring the limit of logarithmic functions, particularly in cases that involve indeterminate forms and the application of L'H?pital's rule. In this article, we will delve into the detailed solution of determining the value of the limit:

Understanding the Limit

Consider the limit:

$$lim_{x to 0} log_{sin x} x$$

Initially, we can express the logarithm with a different base using the natural logarithm:

$$lim_{x to 0} log_{sin x} x lim_{x to 0} frac{ln x}{ln sin x}$$

This is due to the change-of-base formula:

$$log_b a frac{ln a}{ln b}$$

Applying L'H?pital's Rule

As (x) approaches 0, the expression inside the logarithm, (sin x), also approaches 0. This results in an indeterminate form of (frac{infty}{infty}). Hence, the limit can be evaluated using L'H?pital's rule. Applying this rule, we get:

$$lim_{x to 0} frac{ln x}{ln sin x} lim_{x to 0} frac{frac{d}{dx} ln x}{frac{d}{dx} ln sin x}$$

Next, we differentiate both the numerator and the denominator:

$$lim_{x to 0} frac{frac{1}{x}}{frac{1}{sin x} cos x} lim_{x to 0} frac{sin x}{x cos x}$$

Further Simplification using L'H?pital's Rule

Since we still have an indeterminate form of (frac{0}{0}) as (x) approaches 0, we can apply L'H?pital's rule again:

$$lim_{x to 0} frac{sin x}{x cos x} lim_{x to 0} frac{frac{d}{dx} sin x}{frac{d}{dx} (x cos x)}$$

Performing the differentiation, we get:

$$lim_{x to 0} frac{cos x}{- x sin x cos x}$$

Evaluating this limit as (x) approaches 0, we find:

$$lim_{x to 0} frac{cos x}{- x sin x cos x} frac{1}{0 1} 1$$

Hence, the limit is:

$$boxed{lim_{x to 0} log_{sin x} x 1}$$

Conclusion

The limit of the logarithmic function as described in this problem does indeed exist and equals 1. This exploration has provided a clear understanding of how to handle limits with logarithmic bases and the application of L'H?pital's rule in situations involving indeterminate forms.

Key Takeaways: Indeterminate forms, L'H?pital's rule, and the change-of-base formula are essential tools in calculus for evaluating such limits.