Understanding the Limit of ( e^{frac{1}{h}} ) as ( h ) Approaches Zero

When dealing with limits involving exponential functions, we often encounter expressions that can be quite intricate. One such expression is ( e^{frac{1}{h}} ) as ( h ) approaches zero. Understanding this limit requires a careful application of calculus principles. This article will explore the underlying mathematics and provide a step-by-step solution to the problem.

Introduction to the Limit

The given expression is ( e^{frac{1}{h}} ) for ( 0

The Correct Approach

The original expression was incorrectly stated. The correct form should be:

[ lim_{h to 0} frac{e^h - 1}{h} cdot frac{1}{h} cdot e^{-frac{1}{h}} ]

This expression helps us understand the behavior of ( e^{frac{1}{h}} ) as ( h ) approaches zero. Let's break it down step by step.

Expanding ( e^h - 1 )

The Taylor series expansion of ( e^h ) around ( h 0 ) is:

[ e^h 1 h frac{h^2}{2!} o(h^2) ]

Subtracting 1 from both sides, we get:

[ e^h - 1 h frac{h^2}{2!} o(h^2) ]

The leading term is ( h ), which allows us to simplify the numerator:

[ frac{e^h - 1}{h} 1 frac{h}{2!} o(h) approx 1 frac{h}{2} ]

Thus, the numerator simplifies to:

[ frac{e^h - 1}{h} - 1 approx frac{h}{2} ]

Logarithmic Transformation

Next, we consider the logarithm of the exponent:

[ log(frac{1}{h^{frac{1}{h}}}) frac{log(frac{1}{h})}{h} frac{-log(h)}{h} ]

Using the Taylor series expansion of ( log(h) ) around ( 0 ), we get:

[ log(h) -1 frac{1}{2}h - frac{h^2}{3} o(h^2) ]

Substituting this into the expression for the logarithm:

[ frac{-log(h)}{h} frac{1 - frac{1}{2}h frac{h^2}{3} o(h^2)}{h} frac{1}{h} - frac{1}{2} frac{h}{3} o(h) ]

The leading term is ( frac{1}{h} - frac{1}{2} ), which simplifies the exponent:

[ frac{1}{h^{frac{1}{h}}} e^{frac{1}{h} - frac{1}{2}} e^{frac{1}{h}} cdot e^{-frac{1}{2}} ]

Combining the Results

Now, combining the simplified numerator and the simplified exponent, we get:

[ lim_{h to 0} frac{e^h - 1}{h} cdot frac{1}{h} cdot e^{-frac{1}{h}} lim_{h to 0} left(1 frac{h}{2} o(h)right) cdot left(1 frac{1}{2h} - frac{1}{2} o(1)right) cdot e^{-frac{1}{2}} ]

As ( h ) approaches zero, the leading terms dominate:

[ lim_{h to 0} left(1 frac{h}{2} o(h)right) cdot left(1 frac{1}{2h} - frac{1}{2} o(1)right) cdot e^{-frac{1}{2}} frac{1}{e} ]

Therefore, the limit of ( e^{frac{1}{h}} ) as ( h ) approaches zero is ( frac{1}{e} ).

Conclusion

The limit of ( e^{frac{1}{h}} ) as ( h ) approaches zero is ( frac{1}{e} ). This result is derived through careful application of Taylor series expansions and logarithmic transformations. Understanding such limits is crucial for advanced calculus and mathematical analysis.

Keywords

limit of e, exponential function, calculus, mathematical limit, infinity