Problem 69 - Calculus Early Transcendentals: Pearson New International Edition

Question

Begin by plotting the function in an appropriate window. Your computer may indicate that some of these limits do not exist, but, if so, you should be able to interpret the answer as either \(\infty\) or \(-\infty\). \(\lim _{x \rightarrow 0^{+}}(1+\sqrt{x})^{1 / \sqrt{x}}\)

Step-by-Step Solution

Answer

The limit is \(e\).

Step 1: Understand the Function and Limit

We are given the function \((1+\sqrt{x})^{1/\sqrt{x}}\) and asked to find the limit as \(x\) approaches \(0^{+}\), which means from the positive side. Let's rewrite the expression to understand its behavior.

Step 2: Rewrite the Expression Using Exponentials

Recognize that the expression \((1+\sqrt{x})^{1/\sqrt{x}}\) can be rewritten in terms of the exponential function: \[\exp\left(\frac{\ln(1+\sqrt{x})}{\sqrt{x}}\right)\]. This will help us analyze the behavior of the function as \(x\) approaches 0 from the positive side.

Step 3: Analyze the Logarithmic Component

Consider the expression \(\ln(1+\sqrt{x})\). Using the Taylor expansion for small \(\sqrt{x}\), we have \(\ln(1+\sqrt{x}) \approx \sqrt{x} - \frac{x}{2}\). Substituting into our expression gives \[\exp\left(\frac{\sqrt{x} - \frac{x}{2}}{\sqrt{x}}\right) = \exp\left(1 - \frac{\sqrt{x}}{2}\right)\].

Step 4: Evaluate the Limit

Now, evaluate the limit of \(\exp\left(1 - \frac{\sqrt{x}}{2}\right)\) as \(x\) approaches 0 from the positive direction. As \(x \to 0^{+}\), \(\frac{\sqrt{x}}{2} \to 0\), thus the expression simplifies to \(\exp(1) = e\).

Step 5: Conclusion

The analysis shows that the limit of the function \((1+\sqrt{x})^{1/\sqrt{x}}\) as \(x\) approaches 0 from the positive side is \(e\). Hence, the limit exists and is equal to \(e\).

Key Concepts

Exponential Function

Exponential functions are mathematical expressions where variables appear in the exponent. The most common exponential function you'll come across is the natural exponential function, denoted as \( e^x \), where \( e \approx 2.71828 \) is the base of natural logarithms. Exponential functions are crucial in calculus due to their unique growth properties.

• **Rapid Growth**: Exponential functions can grow extremely fast, which is why they're used to model processes like population growth and radioactive decay.
• **Continuous Growth**: They represent continuous compounded growth, distinguishing them from linear or polynomial growth.
• **Behavior at Infinity**: As the exponent increases, the value of an exponential function can become very large, approaching infinity for positive exponents.

In limits, exponential functions help to convert complex forms into something manageable by leveraging their properties and behavior as variables approach certain points.

Taylor Expansion

The Taylor expansion is a powerful tool in calculus, allowing us to approximate complex functions using polynomials. This is particularly helpful when analyzing functions at points that may otherwise be difficult to interpret.

• **Definition**: Essentially, the Taylor expansion represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point.
• **Simplification**: For small values of a variable, like \( \sqrt{x} \) in our example, the Taylor expansion can simplify expressions like \( \ln(1 + \sqrt{x}) \).
• **Example**: For the logarithmic function \( \ln(1 + x) \), the Taylor series expansion around 0 becomes \( x - \frac{x^2}{2} + \frac{x^3}{3} - \ldots \).

Using Taylor expansion in our problem helps approximate \( \ln(1 + \sqrt{x}) \) as \( \sqrt{x} - \frac{x}{2} \), making it easier to evaluate the limit.

Logarithmic Function

Logarithmic functions are the inverse of exponential functions and are represented as \( \ln(x) \) for natural logs or \( \log(x) \) for other bases. They are incredibly useful in transforming multiplicative processes into additive ones, simplifying complex calculations.
Some of their core properties include:

• **Inverse Relationship**: The logarithm of a number is the power to which the base must be raised to yield that number.
• **Derivative**: The derivative of the natural logarithm \( \ln(x) \) is \( \frac{1}{x} \), highlighting its decreasing growth rate as x increases.
• **Rule with Limits**: The logarithmic function can simplify expressions involving exponents, making it easier to analyze limits.

In the provided exercise, taking the natural logarithm \( \ln(1 + \sqrt{x}) \) aids in breaking down the original expression for easy limit evaluation.

Limit Evaluation

Limit evaluation is a fundamental concept in calculus, focusing on what the value of a function approaches as the input nears a specific point. This concept is pivotal when determining the behavior of functions that may not be explicitly defined at that point.

• **Approaching Values**: Limits deal with values that a function approaches, rather than just values that a function is defined to equal.
• **Infinite Limits**: Sometimes, limits can head towards infinity, indicating a function growing without bound as it nears a particular input.
• **Existence of Limits**: A limit exists if the left-hand and right-hand limits as a variable approach a point are equal.

In our exercise, the limit of \( (1+\sqrt{x})^{1/\sqrt{x}} \) as \( x \) approaches 0 from the positive side is evaluated by rewriting the expression using known expansions and properties, ultimately showing it converges to \( e \). This highlights the use of calculus techniques to solve for variables at challenging points.