Problem 58 - University Calculus: Early Transcendentals

Question

Find the limits. $$\lim _{x \rightarrow 0}\left(e^{x}+x\right)^{1 / x}$$

Step-by-Step Solution

Answer

The limit is \(e^2\).

Step 1: Understand the Problem

We need to determine the limit of the expression \((e^x + x)^{1/x}\) as \(x\) approaches 0.

Step 2: Apply Logarithms

To simplify the expression, we take the natural logarithm: \(\ln L = \lim_{x \to 0} \frac{\ln(e^x + x)}{x}\), where \(L\) is the limit we seek.

Step 3: Use L'Hopital's Rule

Since the expression inside the limit \(\frac{\ln(e^x + x)}{x}\) is in the indeterminate form \(\frac{0}{0}\), use L'Hopital's Rule: differentiate the numerator and the denominator.

Step 4: Find Derivatives

Differentiate the numerator: \(\frac{d}{dx}[\ln(e^x + x)] = \frac{e^x + 1}{e^x + x}\). The derivative of the denominator \(x\) is 1.

Step 5: Apply L'Hopital's Rule

Using L'Hopital's Rule: \(\ln L = \lim_{x \to 0} \frac{e^x + 1}{e^x + x}\). Evaluate this limit at \(x = 0\): \(\frac{1 + 1}{1 + 0} = 2\).

Step 6: Solve for L

Since \(\ln L = 2\), exponentiate both sides to find \(L\): \(L = e^2\).

Step 7: Verify the Solution

Checking the calculations, all steps and the application of L'Hopital's Rule are correct. Hence, the solution is verified.

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a valuable tool in calculus for solving limits that result in indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). This rule can often simplify the process of finding limits by allowing us to differentiate the numerator and the denominator instead of directly evaluating the original limit expression. When applying L'Hôpital's Rule, it is important to:

• Ensure that the limit results in an indeterminate form before attempting to apply the rule.
• Differentiating both the numerator and the denominator separately, not forgetting the rules of differentiation.
• Re-evaluating the limit after applying the differentiation to see if the indeterminate form is resolved.

In the given problem, \(\lim_{x \to 0} \frac{\ln(e^x + x)}{x}\), we are dealing with \(\frac{0}{0}\), a classic example where L'Hôpital's Rule can be strategically used to find the limit efficiently. We differentiate the numerator and denominator separately, and then re-evaluate the limit with the new expression formed after differentiation.

Indeterminate Forms

In calculus, indeterminate forms occur in limits that initially do not provide straightforward answers. Common indeterminate forms include \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(\infty - \infty\), \(0 \times \infty\), \(1^\infty\), \(0^0\), and \(\infty^0\).The presence of indeterminate forms suggests:

• The need for further manipulation or transformation of the limit expression.
• The potential use of techniques such as algebraic simplification, factoring, or applying L'Hôpital's Rule.

In the provided exercise, the indeterminate form \(\frac{0}{0}\) arises when trying to solve the limit \(\lim_{x \to 0} \frac{\ln(e^x + x)}{x}\). This happens because both the numerator \(\ln(e^x + x)\) and the denominator \(x\) approach 0 as \(x\) approaches 0. By recognizing the indeterminate form, students can correctly apply advanced calculus tools such as L'Hôpital's Rule to resolve the limit.

Natural Logarithms

Natural logarithms, denoted as \(\ln(x)\), are logarithms with the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. They play a critical role in calculus, especially in solving problems involving exponential functions and limits.Here’s how natural logarithms are useful:

• They help transform complex exponential expressions into simpler additive forms through the logarithm properties.
• They are effective in calculus for differentiating exponential functions, with the derivative of \(\ln(x)\) being \(\frac{1}{x}\).

In the given problem, taking the natural logarithm \( \ln[(e^x + x)^{1/x}] \) transforms the exponentiated form into a more manageable fraction \( \frac{\ln(e^x + x)}{x} \). This step is crucial as it simplifies the process of evaluating the limit using L'Hôpital's Rule and highlights the usefulness of natural logarithms in limit problems of this nature.