Problem 66 - Single Variable Calculus

Question

Evaluate \(\lim _{x \rightarrow 1} \frac{\ln x}{x-1}\). Hint: Use the Taylor series representation of \(\ln x\) at \(1 .\)

Step-by-Step Solution

Answer

Using the Taylor series representation of \(\ln x\) at \(x=1\), we rewrite the given function as: \(\frac{\ln x}{x-1} = 1 - \frac{(x-1)^2}{3} + \frac{(x-1)^4}{5} + \cdots\) Evaluating the limit as \(x\) approaches \(1\): \(\lim _{x \rightarrow 1} \frac{\ln x}{x-1} = \boxed{1}\)

Step 1: Find the Taylor series representation of \(\ln x\) at \(x=1\).

The Taylor series representation of a function \(f(x)\) at \(x=a\) is given by: \[f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots\] For \(f(x) = \ln x\) at \(x=1\), we need to find its derivatives and their values at \(x=1\). \(\frac{d}{dx}\ln x = \frac{1}{x}\) Evaluate the derivative at x=1: \(\frac{1}{1} = 1\) Now consider the second derivative of f(x): \(\frac{d^2}{dx^2}\ln x = \frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2}\) Evaluate the second derivative at x=1: \(-\frac{1}{1^2} = -1\) Considering that the function \(\ln x\) is odd around \(x=1\), only odd-degree terms will appear in the Taylor series expansion. Thus, the Taylor series representation of \(\ln x\) at \(x=1\) is: \(\ln x = (x-1) - \frac{(x-1)^3}{3} + \frac{(x-1)^5}{5} + \cdots\)

Step 2: Use the Taylor series representation to rewrite and simplify the given function.

Using the Taylor series representation obtained in Step 1, we can rewrite the function as: \(\frac{\ln x}{x-1} = \frac{(x-1) - \frac{(x-1)^3}{3} + \frac{(x-1)^5}{5} + \cdots}{x-1}\) Now, divide each term in the numerator by \((x-1)\): \(\frac{\ln x}{x-1} = 1 - \frac{(x-1)^2}{3} + \frac{(x-1)^4}{5} + \cdots\)

Step 3: Evaluate the limit using the simplified function.

Now we can evaluate the limit as \(x\rightarrow1\): \(\lim _{x \rightarrow 1} \frac{\ln x}{x-1} = \lim _{x \rightarrow 1}\left(1 - \frac{(x-1)^2}{3} + \frac{(x-1)^4}{5} + \cdots\right)\) As \(x\) approaches \(1\), all the terms with a power of \((x-1)\) will approach \(0\). Therefore, the limit is equal to: \(\lim _{x \rightarrow 1} \frac{\ln x}{x-1} = 1\) So, the limit of the given function as \(x\rightarrow1\) is \(\boxed{1}\).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828. It has distinct properties that make it a fundamental component in calculus and higher mathematics.
The natural logarithm has several significant characteristics:

• It is the inverse of the exponential function \( e^x \).
• It describes the time needed to reach a certain level of continuous growth.
• Its derivative is \( \frac{1}{x} \), which is vital for understanding logarithmic growth rates.

These properties make the natural logarithm particularly important in solving problems involving exponential growth and decay processes.

Calculus Limit

In calculus, a limit is a fundamental concept that refers to the value that a function approaches as the input (or index) approaches some value. The notation \( \lim_{x \to a} f(x) = L \) implies that as \( x \) gets closer and closer to \( a \), \( f(x) \) approaches \( L \).
Limits are crucial for defining many fundamental concepts in calculus such as continuity, derivatives, and integrals. They help us understand the behavior of functions that might not be immediately apparent from the function's equation. For example, the limit \( \lim_{x \to 1} \frac{\ln x}{x-1} \) evaluates to 1 through simplifying the function with a Taylor series expansion, helping to resolve any ambiguity in division by a form like \( 0/0 \).
This concept of limits also leads to L'Hôpital's Rule, which is often used to find limits involving indeterminate forms.

Derivatives

Derivatives represent the rate of change of a function with respect to a variable. They are central to the study of calculus and have a wide range of applications in fields as diverse as physics, engineering, and economics.
The derivative of a function \( f(x) \) is denoted as \( f'(x) \) or \( \frac{df}{dx} \). For the natural logarithm function \( \ln x \), the derivative is \( \frac{1}{x} \). This indicates how rapidly the logarithm is changing at any point \( x \).
Derivatives can tell you how a change in one variable affects another, enabling the prediction and analysis of different phenomena, from the trajectory of a projectile to the optimization of a production process.

Higher-Order Derivatives

Higher-order derivatives are the derivatives of a derivative, reflecting the rate at which the rate of change itself is changing. The notation for the second derivative of a function \( f(x) \) is \( f''(x) \) or \( \frac{d^2 f}{dx^2} \), and similarly for higher orders.
These are particularly useful in analyzing the behavior of functions beyond initial rates of change, offering insights into the concavity and possible points of inflection. For example, when dealing with the Taylor series expansion of \( \ln x \) at \( x=1 \), we look at both the first and second derivatives:

• The first derivative \( \frac{1}{x} \) evaluates to 1 at \( x=1 \).
• The second derivative \( -\frac{1}{x^2} \) evaluates to -1 at \( x=1 \).

These calculations allow us to construct and simplify Taylor series to evaluate limits and understand more complex function behaviors.