Problem 13 - Calculus for Biology and Medicine

Question

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \infty} \frac{\ln (\ln x)}{x} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Verify Indeterminate Form

Before applying L'Hospital's rule, we need to check if the limit is in an indeterminate form. As \( x \to \infty \), both \( \ln (\ln x) \to \infty \) and \( x \to \infty \). Thus, \( \frac{\ln(\ln x)}{x} \) is in the form \( \frac{\infty}{\infty} \), which is an indeterminate form where L'Hospital's rule can be applied.

Step 2: Apply L'Hospital's Rule

L'Hospital's rule states that if \( \lim_{x \to c} \frac{f(x)}{g(x)} \) is in an indeterminate form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), then \( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \). Here, let \( f(x) = \ln(\ln x) \) and \( g(x) = x \). We need to find their derivatives: \[ f'(x) = (\ln(\ln x))' = \frac{1}{\ln x} \cdot \frac{1}{x} = \frac{1}{x \ln x} \] \[ g'(x) = (x)' = 1 \] Then the limit becomes:\[ \lim_{x \to \infty} \frac{\ln(\ln x)}{x} = \lim_{x \to \infty} \frac{\frac{1}{x \ln x}}{1} \] which simplifies to\[ \lim_{x \to \infty} \frac{1}{x \ln x} \].

Step 3: Evaluate the Simplified Limit

Now we evaluate the limit \( \lim_{x \to \infty} \frac{1}{x \ln x} \). As \( x \to \infty \), both \( x \) and \( \ln x \to \infty \), so \( x \ln x \to \infty \) as well. Therefore, \( \frac{1}{x \ln x} \to 0 \).

Step 4: State the Final Result

The original limit simplifies to 0 after applying L'Hospital's rule and evaluating the expression.

Key Concepts

Indeterminate Forms

In calculus, an indeterminate form is a mathematical expression that is not clearly defined and can lead to different outcomes when evaluated directly. One common example is the form \( \frac{\infty}{\infty} \), which arises when both the numerator and denominator of a fraction approach infinity. This ambiguity requires a special approach to find limits, such as L'Hospital's Rule. Indeterminate forms are crucial in understanding real-world phenomena where limits do not straightforwardly exist. Other forms like \( \frac{0}{0} \), \( \infty - \infty \), or \( 0 \cdot \infty \) also fall under this category and represent situations where traditional limit techniques might fail to apply directly. The presence of an indeterminate form signals the need for additional tools or techniques to properly evaluate a limit without misleading conclusions.

Calculus Limits

Limits help us understand the behavior of functions as they approach specific points or infinity. In our exercise, the limit \( \lim _{x \rightarrow \infty} \frac{\ln (\ln x)}{x} \) is studied as \( x \) approaches infinity. This indicates how the function behaves for extremely large values of \( x \). Limits are foundational in calculus as they are used to define concepts such as continuity, derivatives, and integrals. By evaluating limits, one can infer the value a function tends toward, despite never actually reaching that value.

• The limit \( \frac{c}{\infty} = 0 \) is useful in simplifying expressions.
• Understanding limits aids in the comprehension of series convergence, a key part of analysis.

Limits help tie together the properties of a function at discrete points with its overall behavior.

Derivatives

A derivative represents the rate at which a function changes at any point and is a core concept in calculus. In the context of L'Hospital's Rule, derivatives are used to resolve limits of indeterminate forms by differentiating the numerator and the denominator. The derivative acts like a mathematical lens, allowing us to see how a function grows or shrinks in infinitely small increments. In our example, the derivatives of \( f(x) = \ln(\ln x) \) and \( g(x) = x \) help transform an indeterminate form into a determinable limit \( \frac{1}{x \ln x} \).

• Derivative of a function \( g(x) = c \cdot x^n \) is \( g'(x) = c \cdot nx^{n-1} \).
• Chain rule helps find derivatives of composite functions like \( \ln(\ln x) \).

By taking derivatives, we gain deeper insight into the behavior and eventual outcome of functions, especially as they approach limits.