91Ó°ÊÓ

The natural logarithm, often denoted as \( \ln \), is a logarithm with the base \( e \), where \( e \approx 2.71828 \) is Euler's number. It's a powerful tool for simplifying expressions involving exponents. In problems dealing with limits and growth rates, the natural logarithm helps transform the expression into a more manageable form for analysis.

When you have something like \( n^{1/n} \) as presented in the step-by-step solution, taking the natural logarithm allows us to work with the exponents directly:

• The expression \( \ln(a_n) = \frac{1}{n} \ln n \) is much simpler to evaluate as \( n \) approaches infinity.
• It converts a root into a multiplication by a fraction, enabling the use of calculus tools like L'Hopital's Rule more effectively.

Understanding how to use the natural logarithm will not only simplify expressions with exponents but also assist in solving complex problems involving exponential and logarithmic relationships.

L'Hopital's Rule is a technique in calculus used to find the limit of indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). It states that if the limit of \( \frac{f(x)}{g(x)} \) as \( x \to c \) gives an indeterminate form, then the limit of \( \frac{f'(x)}{g'(x)} \) as \( x \to c \) may provide the solution.

In our exercise, we needed to evaluate \( \lim_{n \to \infty} \frac{\ln n}{n} \). Since both the numerator \( \ln n \) and the denominator \( n \) approach infinity as \( n \to \infty \), L'Hopital's Rule is suitable here. By differentiating:

• The derivative of \( \ln n \) is \( \frac{1}{n} \).
• The derivative of \( n \) is \( 1 \).

Thus, applying L'Hopital's Rule to our limit gives: \[\lim_{n \to \infty} \frac{1/n}{1} = \lim_{n \to \infty} \frac{1}{n} = 0.\]L'Hopital's Rule was essential in transforming our problem into a simpler one that can be solved easily, confirming that the original exponential term approaches zero.

Exponential functions, characterized by the constant base \( e \), are expressions of the form \( e^x \). They play a pivotal role in calculus, especially when dealing with limits, growth, and decay.

After determining that \( \ln(a_n) = \ln(n^{1/n}) \) approaches zero as \( n \to \infty \), we use the properties of exponentials to conclude:

• Since \( \lim_{n \to \infty} \ln(a_n) = 0 \), we can exponentiate both sides. The exponential function \( e^x \) turns zero into \( e^0 = 1 \).
• This step is crucial because it translates a zero logarithm, showing that as \( n \to \infty \), \( a_n \), our original sequence, converges to the value of 1.

The relationship between exponentials and logarithms allowed a complex limit to resolve simply, which is a testament to the power of these mathematical tools in solving problems involving asymptotic behavior of functions.