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When dealing with limits, sometimes we encounter expressions that don't immediately give us clear answers. These are known as "indeterminate forms." An indeterminate form occurs when a limit results in an expression like

• \(0 \cdot \infty\)
• \(\frac{0}{0}\)
• \(\frac{\infty}{\infty}\)

In this exercise, the original expression \(e^{-t} \ln t\) as \(t \to \infty\) results in the form \(0 \cdot \infty\), because \(e^{-t} \to 0\) and \(\ln t \to \infty\). This mix of a zero and infinity is what makes it indeterminate and calls for a method to further analyze and resolve the expression. By rewriting or rearranging the expression, we often transform it into a condition where L'Hôpital's Rule can be helpful.

Evaluating limits involves finding the value that a function approaches as a variable approaches a certain point. The given problem requires us to evaluate the limit \( \lim_{t \rightarrow \infty} e^{-t} \ln t \). Initially, the expression is in an indeterminate form \(0 \cdot \infty\). To evaluate the limit, we need to turn this into a form where we can apply calculus rules effectively.

By strategically rewriting the expression as a fraction, \(\frac{\ln t}{e^t}\), we transform it to the \(\frac{\infty}{\infty}\) indeterminate form, which is a suitable foundation for applying L'Hôpital's Rule. This transformation is crucial because it allows us to use derivatives to simplify and evaluate the limit, leading to clear and understandable results.

The derivative is a fundamental tool in calculus that helps us understand how functions change. When applying L'Hôpital's Rule, we use derivatives of the numerator and the denominator of a fraction.

For the expression \(\frac{\ln t}{e^t}\), we take the derivative of each component:

• The derivative of \(\ln t\) is \(\frac{1}{t}\). This derivative reflects the rate of change of the natural logarithmic function.
• For the denominator, the derivative of \(e^t\) is simply \(e^t\). The exponential function has the special property of being its own derivative.

Thus, using these derivatives, we can simplify the limit expression \(\frac{1/t}{e^t}\) to \(\frac{1}{t e^t}\), which is much easier to evaluate as \(t\) approaches infinity.

Exponential and logarithmic functions are essential in calculus. They're recurring themes in problems involving growth, decay, and logarithmic scales. These functions have unique properties that often come in handy for limit evaluation.

The original expression \(e^{-t} \ln t\) combines these functions:

• The exponential \(e^{-t}\) decreases quickly to zero as \(t\) becomes very large, representing exponential decay.
• The logarithmic function \(\ln t\) increases without bound as \(t\) approaches infinity, representing slow growth.

When analyzing limits, these characteristics can lead to indeterminate forms, like \(0 \cdot \infty\). Understanding the behavior and respective rates of change of exponential and logarithmic functions is critical for rearranging expressions and applying techniques like L'Hôpital's Rule effectively. This knowledge allows us to resolve apparent contradictions, find definite values, and understand the behavior of complex mathematical expressions.