Problem 27 - Calculus Single Variable

Question

Use an algebraic manipulation to put the limit in a form which can be treated using l'Hôpital's Rule; then evaluate the limit. \(\lim _{x \rightarrow \infty} e^{-x} \ln (x)\)

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Recognize the Indeterminate Form

First, we note that as \(x\) approaches infinity, \(e^{-x}\) approaches 0 and \(\ln(x)\) approaches infinity. This suggests an indeterminate form of type \(0 \cdot \infty\). To apply l'Hôpital's Rule, we need a \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) form.

Step 2: Rewrite the Limit in a Suitable Form

Rewrite the expression as \(\lim_{x \to \infty} \frac{\ln(x)}{e^x}\). Now, as \(x\) approaches infinity, the limit becomes a \(\frac{\infty}{\infty}\) form, which is suitable for l'Hôpital's Rule.

Step 3: Apply l'Hôpital's Rule

Differentiate the numerator and the denominator separately. The derivative of \(\ln(x)\) is \(\frac{1}{x}\), and the derivative of \(e^x\) is \(e^x\). Now the limit becomes \(\lim_{x \to \infty} \frac{\frac{1}{x}}{e^x}\).

Step 4: Simplify the Expression

Rewriting the limit, we have \(\lim_{x \to \infty} \frac{1}{x e^x}\). As \(x\) approaches infinity, both \(x\) and \(e^x\) grow, but \(e^x\) grows exponentially faster than \(x\).

Step 5: Evaluate the Final Limit

Since the exponential function outpaces the linear function, \(\frac{1}{x e^x}\) approaches 0 as \(x\) approaches infinity. Thus, the original limit evaluates to 0.

Key Concepts

Understanding Indeterminate Forms

In calculus, an indeterminate form arises when evaluating a limit does not give a clear answer. These forms suggest that more analysis or algebraic manipulation is needed to determine the actual limit. One common indeterminate form is of the type \(0 \cdot \infty\), which implies that one factor is approaching zero while the other is growing without bound. This is exactly what happens when you look at \(e^{-x} \ln(x)\) as \(x\) approaches infinity: \(e^{-x}\) tends to 0 and \(\ln(x)\) tends to infinity.
To handle these situations, we often need to rewrite such expressions to form a ratio that fits \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). These are the specific types of indeterminate forms where l'Hôpital's Rule can be applied effectively. By transforming the expression, you can step towards solving the limit and gaining deeper insights into the behavior of the functions involved.

Exploring Exponential Functions

Exponential functions are a class of functions where the variable appears in the exponent, like \(e^x\). They are among the most powerful and fastest growing functions in mathematics.
When \(x\) takes on very large positive values, \(e^x\) grows rapidly, outpacing not only linear functions like \(x\), but also polynomial functions. This property makes exponential functions especially important in calculus, as they can dominate in expressions involving limits. For example, in the limit \(\lim_{x \rightarrow \infty} \frac{1}{x e^x}\), \(e^x\) grows much faster than \(x\), leading to the entire expression approaching zero.
Another key characteristic of exponential functions is how they behave under differentiation and integration, where they maintain their form, making them unique and versatile in mathematical analysis. Their rapid growth and distinctive properties often require special strategies when working with limits involving them.

The Role of the Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is the logarithm to the base \(e\). It provides the inverse operation of the exponential function \(e^x\). In terms of differentiation, the natural logarithm has a straightforward derivative: \(\frac{d}{dx}\ln(x) = \frac{1}{x}\).
For large values of \(x\), \(\ln(x)\) grows slowly compared to exponential functions. This slow growth rate is what makes exponential functions like \(e^x\) so dominant when compared in limits, such as in our earlier example where \(e^x\) dominates \(\ln(x)\) when both are involved in a function's growth.
When dealing with limits, the natural logarithm often shows up due to its properties that allow us to simplify complex expressions, making it easier to apply rules like l'Hôpital's Rule effectively. These properties make the natural logarithm an essential tool for unraveling the behavior of functions under extreme conditions, such as as \(x\) approaches infinity.