Problem 12 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is 0 as \( x \to \infty \).

Step 1: Understanding the Problem

We are asked to find the limit of \( \lim_{x \to \infty} \frac{\ln x}{\sqrt{x}} \) using l'Hôpital's Rule. This rule is used when we encounter an indeterminate form like \( \frac{\infty}{\infty} \).

Step 2: Determine the Indeterminate Form

As \( x \to \infty \), \( \ln x \to \infty \) and \( \sqrt{x} \to \infty \), so the limit is of the form \( \frac{\infty}{\infty} \). This allows us to apply l'Hôpital's Rule.

Step 3: Differentiate the Numerator and Denominator

By l'Hôpital's Rule, differentiate the numerator \( \ln x \) and the denominator \( \sqrt{x} \). The derivative of \( \ln x \) is \( \frac{1}{x} \), and the derivative of \( \sqrt{x} \) is \( \frac{1}{2\sqrt{x}} \).

Step 4: Apply l'Hôpital's Rule

The limit becomes \( \lim_{x \to \infty} \frac{\frac{1}{x}}{\frac{1}{2\sqrt{x}}} \). This simplifies to \( \lim_{x \to \infty} \frac{2\sqrt{x}}{x} \), which can be further simplified to \( \lim_{x \to \infty} \frac{2}{\sqrt{x}} \).

Step 5: Evaluate the Simplified Limit

As \( x \to \infty \), \( \sqrt{x} \to \infty \), so \( \frac{2}{\sqrt{x}} \to 0 \). Thus, the limit evaluates to 0.

Key Concepts

Limits

Limits are central to calculus and help us understand the behavior of functions as a variable, such as \( x \), approaches a specific value or infinity. When we say \( \lim_{x \to a} f(x) \), we are interested in what value \( f(x) \) tends towards as \( x \) gets closer to \( a \).
When dealing with limits approaching infinity, we often need to simplify complex expressions to find their limiting behavior. For example, \( \lim_{x \to \infty} \frac{\ln x}{\sqrt{x}} \) investigates how the ratio of the natural logarithm of \( x \) to the square root of \( x \) behaves as \( x \) becomes very large. It is crucial for testing convergence or divergence of functions.

Indeterminate Forms

An indeterminate form is a mathematical expression where direct substitution does not readily lead to a conclusion or a definite value. Commonly, this happens when evaluating \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
Indeterminate forms signal that additional work is required, such as using algebraic manipulation or calculus tools. In our exercise, \( \lim_{x \to \infty} \frac{\ln x}{\sqrt{x}} \) initially results in \( \frac{\infty}{\infty} \), which is an indeterminate form. This is our cue to use techniques like l'Hôpital's Rule to evaluate it appropriately, making these forms crucial stepping stones in limit problems.

Differentiation

Differentiation is a fundamental concept in calculus, involving the computation of derivatives, which measure the rate at which a function changes.
For a given function \( f(x) \), its derivative \( f'(x) \) represents the slope of the tangent line at any point \( x \).
In applying l'Hôpital's Rule, we differentiate both the numerator and denominator separately. For \( \ln x \), the derivative is \( \frac{1}{x} \), and for \( \sqrt{x} \), the derivative is \( \frac{1}{2\sqrt{x}} \).
These derivatives help us to transform and simplify the original indeterminate form \( \frac{\infty}{\infty} \) into a solvable expression by exploring the limits of these newly differentiated terms.

Infinity

Infinity is a concept in mathematics that describes something endless or without bound. In calculus, it is used to understand the behavior of functions as their inputs grow very large or very small.
Problems like \( \lim_{x \to \infty} \frac{\ln x}{\sqrt{x}} \) use the idea of infinity to predict how functions behave at extreme values.
When evaluating such limits, we analyze the terms within the function that might grow towards infinity, leading us to apply rules like l'Hôpital's Rule, which is specifically tailored to handle limits that involve infinity, to examine the overall effect of unbounded growth.