Understanding the Limit of ( left( frac{x}{sin x} right)^{6/x^2} ) as ( x ) Approaches 0

In mathematics, evaluating limits can be a complex task, especially when dealing with expressions involving trigonometric functions and exponents. This article will guide you through finding the limit of the expression ( left( frac{x}{sin x} right)^{6/x^2} ) as ( x ) approaches 0. We will break down the solution step by step, ensuring clarity and understanding.

Step 1: Rewrite the Expression

Let us denote the expression as ( y ), where:

( y left( frac{x}{sin x} right)^{6/x^2} )

Step 2: Take the Natural Logarithm

By taking the natural logarithm on both sides, we get:

( ln y frac{6}{x^2} ln left( frac{x}{sin x} right) )

Step 3: Simplify ( ln left( frac{x}{sin x} right) )

To simplify ( ln left( frac{x}{sin x} right) ), we start by using the Taylor series expansion of ( sin x ) around ( x 0 ). Recall:

( sin x x - frac{x^3}{6} O(x^5) )

Thus, we can approximate:

( frac{x}{sin x} frac{1}{1 - frac{x^2}{6} O(x^4)} )

Using the approximation ( frac{1}{1 - u} approx 1 u ) for small ( u ), we have:

( frac{x}{sin x} approx 1 frac{x^2}{6} )

Now, taking the natural logarithm:

( ln left( frac{x}{sin x} right) approx ln left( 1 frac{x^2}{6} right) )

For small ( x ), ( ln left( 1 u right) approx u ), so:

( ln left( frac{x}{sin x} right) approx frac{x^2}{6} )

Step 4: Substitute Back into the Limit

Substitute this result back into the expression for ( ln y ):

( ln y approx frac{6}{x^2} cdot frac{x^2}{6} 1 )

As ( x ) approaches 0, ( ln y ) approaches 1. Therefore:

( y to e^1 e )

Conclusion

Thus, the limit is:

( lim_{x to 0} left( frac{x}{sin x} right)^{frac{6}{x^2}} e )

To summarize, the limit of ( frac{x}{sin x} ) as ( x ) approaches 0 is 1, not 0/0. The expression ( frac{6}{x^2} ) as ( x ) approaches 0 is not an indeterminate form; it tends to positive infinity. The key to solving this problem lies in using the Taylor series expansion and the properties of logarithms and exponents.

Key Points:

Understanding the behavior of trigonometric functions as ( x ) approaches 0 Using Taylor series expansion to find the limit of complex expressions Applying logarithmic and exponential properties to simplify and evaluate the limit