Evaluating the Limit of ( left(frac{x}{sin x}right)^x ) as ( x ) Approaches 0

Understanding the behavior of functions as they approach certain values is a fundamental aspect of calculus. One such interesting case is the evaluation of the limit:

Definition: The limit of a function ( f(x) ) as ( x ) approaches ( a ) is the value that ( f(x) ) gets arbitrarily close to as ( x ) gets closer to ( a ).

Case Analysis and Limits Involving ( frac{x}{sin x} )

In the given expression ( left(frac{x}{sin x}right)^x ), as ( x ) approaches 0, the term ( frac{x}{sin x} ) approaches 1. This is a well-known limit that can be derived from the fact that:

( lim_{x to 0} frac{sin x}{x} 1 )

Simplifying the Expression

Let's rewrite the expression to make it clearer:

( displaystyle lim_{x to 0} left( frac{x}{sin x} right)^x left[ lim_{x to 0} frac{x}{sin x} right]^x )

Given that ( lim_{x to 0} frac{x}{sin x} 1 ), the expression simplifies to:

( left(1right)^x )

This is a standard form where any number (in this case, 1) raised to the power of 0 equals 1:

( 1^0 1 )

Alternative Approach Using Natural Logarithm

Another method to approach this problem is to use the natural logarithm to simplify the expression. Let's define:

( L lim_{x to 0} left( frac{x}{sin x} right)^x )

Then,

( ln L lim_{x to 0} x ln left( frac{x}{sin x} right) )

This can be broken down using properties of logarithms:

( ln L lim_{x to 0} left[ x left( ln x - ln (sin x) right) right] )

Further Simplification

Now, consider the limit:

( lim_{x to 0} ln left( frac{x}{sin x} right) 0 )

This is because:

( ln left( frac{x}{sin x} right) ln x - ln (sin x) )

And as ( x ) approaches 0, both ( ln x ) and ( ln (sin x) ) approach negative infinity, but their difference approaches 0.

Therefore:

( lim_{x to 0} x left( ln x - ln (sin x) right) lim_{x to 0} x cdot 0 0 )

Thus:

( ln L 0 )

Exponentiating both sides, we get:

( L e^0 1 )

Conclusion

To summarize, the limit of ( left( frac{x}{sin x} right)^x ) as ( x ) approaches 0 is:

( lim_{x to 0} left( frac{x}{sin x} right)^x 1 )

This result can be derived using natural logarithms or by recognizing the standard form of the expression.