Problem 32 - Essential Calculus

Question

In Exercises \(27-38,\) (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L'Hôpital's Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}\)

Step-by-Step Solution

Answer

The limit of the function as x approaches infinity is \(e\), which is about \(2.718\).

Step 1: Identify the Indeterminate Form

Plug \(x = \infty\) into the function to identify the form. Thus, \(\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x} = (1 + 0)^{\infty}\), which is an indeterminate form. The form is \(1^{\infty}\) which is indeterminate.

Step 2: Evaluate the limit

Translate the given limit into an easier form by taking natural logarithm. The limit then becomes \(\lim _{x \rightarrow \infty} x \cdot \ln\left(1+\frac{1}{x}\right)\). Changing the variable from \(x\) to \(t = \frac{1}{x}\), the expression and limit transforms to \(\lim _{t \rightarrow 0^{+}} \frac{\ln(1+t)}{t}\). Here L'Hôpital's Rule can be applied because the new form of the limit is \(\frac{0}{0}\). Taking derivatives of numerator and denominator, the limit becomes \(\lim _{t \rightarrow 0^{+}} \frac{1/(1+t)}{1} = 1\). But we need to find out the value of the original limit. Since we took natural logarithm before taking the limit, we need to take exponent of the result. So, \(e^1 = e\) will be the final value of the limit.

Step 3: Verify the Result Graphically

When \(x\) approaches infinity, \(y=(1+ 1/x)^x\) approaches a straight horizontal line \(y=e\). From the graph, it confirms the limit is \(e\).

Key Concepts

Understanding Indeterminate Forms

Indeterminate forms arise when evaluating limits, leading to expressions that seem undefined or ambiguous at first glance. In this exercise, when we attempt to find the limit \( \lim_{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x} \), we directly substitute \( x = \infty \) to get \( (1+0)^{\infty} \). This results in the form \( 1^{\infty} \), which is indeterminate. This form tells us that more analysis is required to evaluate the limit.

There are several types of indeterminate forms, such as \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0 \cdot \infty \), and others. Recognizing these forms is essential because they indicate that straightforward substitution won't work, and techniques like transformation or L'Hôpital's Rule may be necessary.

Evaluating Limits with L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool for evaluating limits that result in indeterminate forms. In the given problem, the transformation involving natural logarithms simplifies our evaluation.

To handle the \( 1^{\infty} \) form, we take the natural logarithm, turning the problem into \( \lim_{x \rightarrow \infty} x \cdot \ln\left(1+\frac{1}{x}\right) \). By changing the variable to \( t = \frac{1}{x} \), we analyze \( \lim_{t \rightarrow 0^{+}} \frac{\ln(1+t)}{t} \). This leads to a \( \frac{0}{0} \) form suitable for L'Hôpital's Rule.

Applying the rule involves differentiating the numerator and denominator, simplifying the limit to \( \lim_{t \rightarrow 0^{+}} \frac{1/(1+t)}{1} = 1 \). Thus, the original limit becomes \( e^{1} = e \). Using L'Hôpital's Rule allows us to unravel complex expressions into solvable limits.

Verifying Results with Graphing Techniques

Graphing techniques offer visual verification of our algebraic work. After calculating the limit algebraically, it's helpful to use graphing to confirm the result. In this exercise, graphing \( y = \left(1+\frac{1}{x}\right)^{x} \) provides a visual check on our evaluation.

As \( x \) approaches infinity, the graph of this function tends toward a horizontal line at \( y = e \). This visual confirmation supports our finding that the limit is indeed \( e \).

Graphing can highlight asymptotic behaviors, convergence to particular values, or potential errors in computation. It provides an intuitive grasp of mathematical concepts and complements analytical methods.