Problem 21 - Calculus for Biology and Medicine

Question

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow \infty} \frac{(\ln x)^{2}}{x^{2}} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Form of the Limit

The limit \( \lim_{x \to \infty} \frac{(\ln x)^2}{x^2} \) is presented. As \(x\) approaches infinity, both the numerator \((\ln x)^2\) and denominator \(x^2\) approach infinity. This is an \(\frac{\infty}{\infty}\) indeterminate form, suggesting the potential use of l'Hôpital's Rule.

Step 2: Apply l'Hôpital's Rule

Since the limit is in the \(\frac{\infty}{\infty}\) form, we can apply l'Hôpital's Rule. Differentiate the numerator and the denominator separately: - Derivative of the numerator: \(\frac{d}{dx}(\ln x)^2 = 2(\ln x) \cdot \frac{1}{x} = \frac{2\ln x}{x}\)- Derivative of the denominator: \(\frac{d}{dx}(x^2) = 2x\)Applying l'Hôpital's Rule, the limit becomes:\[ \lim_{x \to \infty} \frac{\frac{2\ln x}{x}}{2x} = \lim_{x \to \infty} \frac{2\ln x}{2x^2} = \lim_{x \to \infty} \frac{\ln x}{x^2} \]

Step 3: Reapply l'Hôpital's Rule if Necessary

The new limit \( \lim_{x \to \infty} \frac{\ln x}{x^2} \) is again an \(\frac{\infty}{\infty}\) form.- Derivative of the numerator: \(\frac{d}{dx} \ln x = \frac{1}{x}\)- Derivative of the denominator: \(\frac{d}{dx} (x^2) = 2x\)Reapply l'Hôpital's Rule:\[ \lim_{x \to \infty} \frac{\frac{1}{x}}{2x} = \lim_{x \to \infty} \frac{1}{2x^2} \]

Step 4: Evaluate the Final Limit

The expression \( \lim_{x \to \infty} \frac{1}{2x^2} \) needs to be evaluated. As \(x\) approaches infinity, the denominator \(2x^2\) becomes very large, causing the whole fraction to approach zero. Thus, the limit is:\[ 0 \]

Key Concepts

Understanding Limits in Calculus

In calculus, a limit describes the value that a function approaches as the input approaches some point. It's like getting closer and closer to a destination but never really arriving.
For instance, in the problem of finding \( \lim_{x \rightarrow \infty} \frac{(\ln x)^{2}}{x^{2}} \), we want to know what value the function approaches as \(x\) becomes infinitely large.
Limits help us understand the behavior of functions at points where they aren't defined or as inputs grow very large or very small.

• They are foundational in defining derivatives and integrals.
• They handle scenarios where straightforward computation is impossible.

As we analyze the limit given in the exercise, we see it leads to a special form known as an indeterminate form.

Identifying and Handling Indeterminate Forms

Indeterminate forms arise in calculus when we have expressions like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), where it's unclear what the limit should be. They signal the need for more refined techniques.
When dealing with \( \lim_{x \rightarrow \infty} \frac{(\ln x)^2}{x^2} \), you notice both parts grow without bound. This is a classic \( \frac{\infty}{\infty} \) form.
To resolve such cases, we can use l'Hôpital's Rule, which provides a way to evaluate these limits by differentiating the numerator and the denominator.

• First, differentiate the top and bottom separately.
• Apply l'Hôpital's Rule repeatedly if another indeterminate form emerges.
• Eventually, the expression simplifies to a determinate form, making the limit clear.

Understanding and working with indeterminate forms is crucial for mastering higher-level calculus.

Applying Calculus Concepts and l'Hôpital's Rule

In calculus, l'Hôpital's Rule is a powerful tool for evaluating limits that result in indeterminate forms such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). It allows us to transform complex limits into simpler ones by differentiating the numerator and the denominator.
Applying this rule to \( \lim_{x \to \infty} \frac{(\ln x)^{2}}{x^{2}} \) involves several steps:

• First, differentiate \((\ln x)^2\) to get \( \frac{2\ln x}{x} \).
• Then, differentiate \(x^2\) to get \(2x\).
• Keep applying l'Hôpital's Rule until the expression is no longer indeterminate.

In our exercise, after reapplying the rule, we eventually simplify to \( \frac{1}{2x^2} \), which clearly approaches zero as \(x\) goes to infinity.
By mastering concepts like l'Hôpital's Rule, students can tackle complex calculus problems with confidence.