Introduction to the Limit of a Complex Function

The analysis of limits can often pose challenging mathematical questions, particularly when dealing with complex functions. This article provides a detailed explanation of how to find the limit of a function as it approaches a certain value, using a case study involving the expression [lim_{x to 0} left(frac{sin x}{x}right)^{frac{1}{x^2}}].

Understanding the Expression

[lim_{x to 0} left(frac{sin x}{x}right)^{frac{1}{x^2}}] is a complex expression involving trigonometric and exponential functions. To approach this problem, we start by taking the natural logarithm of the expression, which simplifies the exponentiation and allows the use of L'Hopital's Rule to resolve the indeterminate form.

Using the Natural Logarithm

Let [ln L lim_{x to 0} ln left[left(frac{sin x}{x}right)^{frac{1}{x^2}}right]]. By the properties of logarithms, we have [ln L lim_{x to 0} frac{ln left(frac{sin x}{x}right)}{x^2}].

Indeterminate Form L'Hopital's Rule

The expression inside the limit is of the indeterminate form [frac{0}{0}], indicating that we need to apply L'Hopital's Rule. L'Hopital's Rule is a method of evaluating limits that involves differentiating the numerator and denominator separately.

Applying L'Hopital's Rule

First differentiation yields: [ln L lim_{x to 0} frac{frac{1}{sin x} cdot cos x cdot x^2 - 2x ln left(frac{sin x}{x}right)}{2x^3}].

Further simplification leads to: [ln L lim_{x to 0} frac{cos x cdot x - sin x}{2x^2 sin x}].

Applying L'Hopital's Rule again, we get: [ln L lim_{x to 0} frac{-sin x cdot x - cos x}{2x sin x 2x^2 cos x - 2x cos x - 2x^2 sin x}].

This simplifies to: [ln L lim_{x to 0} frac{-sin x cdot x}{2x sin x}].

Finally, applying the notable limit (lim_{x to 0} frac{sin x}{x} 1), we find:

[ln L -frac{1}{6}].

Solving for L

By exponentiating both sides, we obtain the final result:

[L e^{-frac{1}{6}}].

Conclusion

The exploration of the limit [lim_{x to 0} left(frac{sin x}{x}right)^{frac{1}{x^2}}] demonstrates the power of L'Hopital's Rule in resolving complex indeterminate forms. Understanding such mathematical concepts is crucial for students and professionals in fields such as engineering, physics, and computer science, where precise mathematical analysis is essential.