91Ó°ÊÓ

In calculus, indeterminate forms often arise when trying to evaluate limits. These forms, such as \(0/0\), \(\infty/\infty\), or \(\infty^0\), signify an expression where direct substitution doesn't easily yield a determinable limit.

Recognizing these forms is crucial as they require specific techniques, like L'Hôpital's Rule or algebraic manipulation, to resolve. In our exercise, the expression \(x^{1/\ln x}\) results in the indeterminate form \(\infty^0\) as \(x\) approaches infinity.

The base \(x\) heads towards infinity and the exponent \(\frac{1}{\ln x}\) heads towards zero, leading to this indeterminacy. Such forms hint that a transformation or rethinking is necessary to find the limit.

Logarithmic transformation can simplify complex limit expressions, especially those involving powers. It employs the properties of logarithms to conditionally restructure and later solve the expression.

For the expression \(x^{1/\ln x}\), taking the natural logarithm helps by converting the problem of exponential growth into one of linear growth. The transformation is as follows:

• Set the expression equal to a variable, say \(y = x^{1/\ln x}\).
• Take the natural log of both sides: \(\ln y = \ln(x^{1/\ln x})\).
• Using properties of logarithms, simplify to \(\ln y = \frac{1}{\ln x} \cdot \ln x = 1\).

This simplification reveals that the logarithm of \(y\) is constant and equals 1.

Therefore, logarithmic transformation resolves the indeterminate form and props us up to solve the limit.

Exponential functions, characterized by their rapid growth or decay, often emerge in calculus while resolving limits, derivatives, and integrals.

In our problem, after transforming the expression using logs, we reached \(\ln y = 1\). To return to the function of \(y\), we exponentiate both sides of the equation. This helps to solve for the original expression:

• From \(\ln y = 1\), you take the exponential of both sides: \(y = e^1\).
• This simplifies to \(y = e\), a constant value, as \(x\) approaches infinity.

Thus, the original expression \(x^{1/\ln x}\) simplifies and converges to \(e\).

Utilizing exponential functions makes solving limits straightforward once the expression has been simplified by other means, like logarithmic transformation.