Question

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \infty} \frac{\ln \left(\ln x^{1000}\right)}{\ln x} $$

Short answer

0

Step-by-step solution

Simplify the Expression

Begin by simplifying the expression inside the logarithm. Notice that \( x^{1000} = e^{1000 \ln x} \). Then, \( \ln(x^{1000}) = 1000 \ln x \). Thus, the expression becomes \( \frac{\ln(1000 \ln x)}{\ln x} \).

Verify Indeterminate Form

As \( x \to \infty \), both the numerator \( \ln(1000 \ln x) \) and the denominator \( \ln x \) approach infinity, leading to the indeterminate form \( \frac{\infty}{\infty} \). This allows us to use l'Hôpital's Rule.

Apply l'Hôpital's Rule

Apply l'Hôpital's Rule, which involves differentiating the numerator and the denominator separately. The derivative of the numerator \( \ln(1000 \ln x) \) is \( \frac{1}{1000 \ln x} \cdot \frac{d}{dx}(1000 \ln x) = \frac{1}{\ln x} \cdot \frac{1}{x} \cdot 1000 \), resulting in \( \frac{1000}{x \ln x} \). The derivative of the denominator \( \ln x \) is \( \frac{1}{x} \).

Simplify the New Expression

Rewrite the limit expression after differentiation as follows: \[\lim_{x \to \infty} \frac{\frac{1000}{x \ln x}}{\frac{1}{x}} = \lim_{x \to \infty} \frac{1000}{\ln x}.\] Simplifying this gives \( \frac{1000}{\ln x} \).

Evaluate the Limit

Finally, as \( x \to \infty \), \( \ln x \to \infty \) as well. Hence, \( \frac{1000}{\ln x} \to 0 \). Thus, \( \lim_{x \to \infty} \frac{\ln(\ln x^{1000})}{\ln x} = 0 \).

Key concepts

Indeterminate Forms

When working with limits in calculus, you may encounter expressions that seem to be impossible to solve at first glance. These are known as indeterminate forms. Indeterminate forms occur when you have a mathematical expression where direct substitution does not give a clear result. For example, expressions like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) are considered indeterminate because plugging values directly into these forms does not yield a specific number or answer.

In this exercise, as \( x \to \infty \), the expressions \( \ln(1000 \ln x) \) and \( \ln x \) both approach infinity, resulting in the indeterminate form \( \frac{\infty}{\infty} \). Recognizing this form is crucial because it allows us to apply advanced calculus tools like l'Hôpital's Rule to investigate the limit further. Understanding these forms is key to unraveling problems involving limits in calculus.

Limits at Infinity

Limits at infinity deal with evaluating the behavior of functions as the variable \( x \) goes towards infinity or negative infinity. In simpler terms, it's about seeing what happens to the outputs of a function as the input gets very large, either in the positive or negative direction.

In the provided exercise, we analyze the limit of the function \( \frac{\ln(\ln x^{1000})}{\ln x} \) as \( x \to \infty \). This involves understanding how the logarithmic functions grow, as they tend to increase slowly compared to polynomial or exponential functions. Understanding the behavior of functions at infinity helps us make predictions about their long-term behavior and supports calculations in higher-level calculus.

Calculus Differentiation

Calculus differentiation is a process of finding the rate at which a function is changing at any given point. It's the backbone of calculus, used to solve problems involving tangent lines, velocity, and optimization, among others.

In this exercise, differentiation comes into play through l'Hôpital's Rule, which necessitates differentiating the numerator \( \ln(1000 \ln x) \) and the denominator \( \ln x \). By applying the chain rule for differentiation, the derivative of the numerator becomes \( \frac{1000}{x \ln x} \), while the derivative of the denominator is \( \frac{1}{x} \). Successfully differentiating these expressions allows us to simplify the original limit problem into a more manageable form, ultimately leading to the solution.

Logarithmic Functions

Logarithmic functions, often simplified as logs, help us understand the inverse relationship of exponential functions. Their unique ability to transform multiplicative relationships into additive ones is crucial in many areas, like calculating compound interest or understanding sound intensity levels.

In the context of this exercise, logarithms are pivotal in simplifying the expression \( \ln(x^{1000}) \), as it transforms to \( 1000 \ln x \). This transformation is crucial because it shapes how the expression behaves as \( x \to \infty \). Understanding the properties and applications of logarithmic functions allows us to navigate and solve complex problems like the one presented here more effectively.