Understanding the Limit of the Logarithm of Tangent Functions as x Approaches 0

Introduction: In this article, we explore the limit of a complex logarithmic function involving the tangent function. Specifically, we evaluate the limit as ( x ) approaches 0 of the logarithm of ( tan(2x) ) with base ( tan(x) ). This problem introduces a series of mathematical techniques, including the change of base formula and L'H?pital's Rule. We will provide a detailed step-by-step solution to this intriguing problem.

Change of Base Formula and Initial Setup

The first step in solving the limit is to transform the expression using the change of base formula for logarithms:

[ log_{tan x} tan(2x) frac{log tan(2x)}{log tan x} ]

Our goal now is to evaluate the following limit:

[ lim_{x to 0} frac{log tan(2x)}{log tan x} ]

Both ( tan(2x) ) and ( tan(x) ) approach 0 as ( x ) approaches 0. Consequently, the logarithms of these values approach negative infinity. This prompts the application of L'H?pital's Rule, which addresses indeterminate forms of the type ( frac{-infty}{-infty} ).

Applying L'H?pital's Rule

First, we differentiate the numerator and the denominator of the fraction. Let:

( f(x) log tan(2x) ) ( g(x) log tan(x) )

The derivatives are:

Numerator: ( f'(x) frac{2 sec^2(2x)}{tan(2x)} cdot 2 frac{4 sec^2(2x)}{tan(2x)} ) Denominator: ( g'(x) frac{sec^2(x)}{tan(x)} )

Applying L'H?pital's Rule, we get:

[ lim_{x to 0} frac{frac{4 sec^2(2x)}{tan(2x)}}{frac{sec^2(x)}{tan(x)}} lim_{x to 0} frac{4 sec^2(2x) tan(x)}{sec^2(x) tan(2x)} ]

Evaluating the Limit as x Approaches 0

For values of ( x ) close to 0, we can use the small-angle approximations:

( tan(x) sim x ) ( tan(2x) sim 2x ) ( sec^2(x) to 1 ) as ( x to 0 ) ( sec^2(2x) to 1 ) as ( x to 0 )

Substituting these approximations, we simplify the expression:

[ lim_{x to 0} frac{4 cdot 1 cdot x}{1 cdot 2x} lim_{x to 0} frac{4x}{2x} lim_{x to 0} 2 2 ]

This confirms the limit evaluates to 2.

Conclusion

The limit of ( log_{tan x} tan(2x) ) as ( x ) approaches 0 is:

[ boxed{2} ]

Alternative Approaches

In addition to the above method, we also explored an alternative approach involving the tangent double-angle identity and logarithm properties. Both methods ultimately lead to the same result:

Using the tangent double-angle identity: Considering the small-angle approximations:

In conclusion, we have demonstrated the use of mathematical tools such as L'H?pital's Rule and the change of base formula to evaluate a complex limit involving logarithmic and trigonometric functions. This problem showcases the importance of careful manipulation and the application of calculus techniques in solving limit problems.