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The natural logarithm is a fundamental mathematical function denoted as \( \ln(x) \). It is the inverse of the exponential function with base \( e \), where \( e \approx 2.71828 \), known as Euler's number. The natural logarithm has the unique property that \( \ln(e^x) = x \) for any real number \( x \).

Using the natural logarithm can simplify problem-solving, especially in calculus. When faced with expressions like \( x^{1/x} \), taking their natural logarithm can transform them into a more manageable form, making it easier to analyze limits and perform differentiation and integration.

- Any logarithm function grows relatively slow compared to polynomial or exponential functions.- For large values of \( x \), \( \ln(x) \) increases, but its rate of increase diminishes, which is crucial when evaluating limits.- The properties of the natural logarithm help us solve limits like \( \lim_{x \rightarrow +\infty} \frac{\ln x}{x} \), which plays a role in exercises such as the one given.

L'Hôpital's Rule is a powerful tool in calculus used to find limits that present an indeterminate form such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). This rule allows us to differentiate the numerator and the denominator separately to simplify the limit calculation.

- It applies when the original function results in an indeterminate form. By differentiating, we often bypass the complexities of algebraic simplification.- For the limit \( \lim_{x \to +\infty} \frac{\ln x}{x} \), since both \( \ln x \) and \( x \) individually approach infinity, the expression \( \frac{\ln x}{x} \) falls into this category.- After differentiating, we get \( \lim_{x \to +\infty} \frac{\ln x}{x} = \lim_{x \to +\infty} \frac{\frac{1}{x}}{1} = \lim_{x \to +\infty} \frac{1}{x} = 0 \), simplifying the limit expression significantly. This illustrates how L'Hôpital's Rule efficiently deals with seemingly complex limits.

The exponential function is represented by \( e^x \), where \( e \) is Euler's number. This function rapidly increases as \( x \) grows larger, showcasing exponential growth.

- In the original problem, we transformed the expression \( x^{1/x} \) into \( y = e^{\frac{1}{x} \ln x} \). This transformation is crucial as it connects the solution to the behavior of exponential functions.- Due to the limit \( \lim_{x \to +\infty} \frac{\ln x}{x} = 0 \), we apply this result to the exponential function: \( e^0 = 1 \).- Such a simplification shows how the exponential function reacts to inputs, leading to the conclusion that \( \lim_{x \to +\infty} x^{1/x} = 1 \).- Understanding this connection highlights the exponential function's role in solving complex limits by turning products or quotients into easier-to-interpret expressions.