Problem 70 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises 69-71, find the limit. Give a convincing argument that the value is correct. $$\lim _{x \rightarrow \infty} \frac{\ln x}{\log x}$$

Step-by-Step Solution

Answer

The limit as x approaches infinity of \( \frac{\ln x}{\log x} \) is \( \ln 10 \).

Step 1: Identifying the form and using L’Hospital’s Rule

The given function is \( \frac{\ln x}{\log x} \) as x tends towards infinity. This is an indeterminate form of the type ∞/∞. Hence, apply L’Hospital’s Rule which states that \(\frac{f(x)}{g(x)} \) as x goes to a value is equal to \( \frac{f'(x)}{g'(x)} \). Apply the rule to the given function. Also remember, \( \frac{d}{dx} (\ln x) = \frac{1}{x} \) and \( \frac{d}{dx} (\log_{10} x) = \frac{1}{x \ln 10} \).

Step 2: Evaluate the derivatives' limit

After differentiating numerator and denominator, the limit becomes \lim_{x \rightarrow \infty} \frac{1/x}{1/(x \ln 10)} = \lim_{x \rightarrow \infty} \frac{\ln 10}{1}. Since \(\ln 10\) is a constant, as \(x\) tends to \(\infty\), the constant remains the same.

Step 3: Final evaluation

The limit of a constant as \(x\) approaches any number (including \(\infty\)) is simply that constant. Hence, \( \lim_{x \rightarrow \infty} \frac{\ln x}{\log x} = \ln 10.\)

Key Concepts

L'Hospital's Rule

L'Hospital's Rule is a powerful tool in calculus for finding the limits of indeterminate forms. When you encounter a limit of the form \( \frac{f(x)}{g(x)} \) where both \( f(x) \) and \( g(x) \) approach 0 or infinity as \( x \) approaches a certain value, L'Hospital's Rule gives us a way to evaluate it.

The rule states that if \( \lim_{x \to c} f(x) = \lim_{x \to c} g(x) = 0 \) or \( \pm \infty \), then:

• \( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \)

provided the limit on the right-hand side exists. This means that instead of calculating the limit of the original functions, you differentiate the numerator \( f(x) \) and the denominator \( g(x) \) first.

You continue applying L'Hospital's Rule until you achieve a determinate form or a recognizable limit. It's a method that simplifies complex limits and is particularly useful when dealing with rational functions.

Indeterminate Forms

In calculus, indeterminate forms arise when the limit you're trying to calculate gives an expression that's not immediately clear. The seven common forms are: 0/0, ∞/∞, 0 × ∞, ∞ - ∞, 0⁰, 1^∞, and ∞⁰.

These forms suggest ambiguity in computation, prompting methods such as L'Hospital's Rule to provide clarity. For instance, when dealing with the form ∞/∞, it isn't obvious if the numerator grows faster, slower, or at the same rate as the denominator. This is where calculus tools step in.

• L'Hospital's Rule is applied specifically to 0/0 and ∞/∞ forms.
• Other forms often require algebraic manipulation or special techniques to resolve.

Recognizing indeterminate forms is vital, as it tells you when further analysis is needed to determine a limit's true value.

Derivative

A derivative, in simple terms, measures how a function changes as its input changes. It's the fundamental concept of calculus, representing an "instantaneous rate of change."

Derivatives can be thought of as the slope of a function at any given point. If \( y = f(x) \), then the derivative \( f'(x) \) describes how \( y \) changes with respect to a small change in \( x \).

• The derivative of \( \ln x \) is \( \frac{1}{x} \).
• The derivative of \( \log_{10} x \) is \( \frac{1}{x \ln 10} \).

Understanding how to calculate derivatives is essential for applying L'Hospital’s Rule, as it involves using these derivatives to resolve limits. Derivatives not only solve limits but also play a key role in understanding the behavior and sketching curves of functions.