Question

In Exercises 2.4.2-2.4.40, find the indicated limits. $$ \lim _{x \rightarrow \infty} \frac{(\log x)^{\beta}}{x} $$

Short answer

Question: Determine the limit of the function \((\log x)^\beta / x\) as \(x\) approaches infinity. Answer: The limit of the given function as \(x\) approaches infinity is 0.

Step-by-step solution

Identifying the indeterminate form

As \(x\) approaches infinity, the log(x) approaches infinity as well. Similarly, \((\log x)^\beta\) approaches infinity, and of course, \(x\) approaches infinity as well. So, we have an "infinity/infinity" indeterminate form.

Apply L'Hopital's Rule

Let's apply L'Hopital's Rule to our limit: $$ \lim_{x\rightarrow\infty}\frac{(\log x)^{\beta}}{x} = \lim_{x\rightarrow\infty}\frac{\mathrm{d}(\log x)^{\beta}}{\mathrm{d}x}\cdot\frac{1}{\frac{\mathrm{d}x}{\mathrm{d}(\log x)}} = \lim_{x\rightarrow\infty}\frac{\beta(\log x)^{\beta-1}}{x} $$ We first differentiate \((\log x)^{\beta}\) with respect to x which is \(\beta(\log x)^{\beta-1}.\) Then we divide this derivative by the derivative of x which is 1. We end up with the same limit expression with an updated exponent: \((\beta - 1)\).

Analyzing the updated limit and applying L'Hopital's Rule again if needed

Now we have the following limit: $$ \lim_{x\rightarrow\infty}\frac{\beta(\log x)^{\beta-1}}{x} $$ Notice that we still have an "infinity/infinity" indeterminate form because \(\beta(\log x)^{\beta-1}\) approaches infinity as \(x\) approaches infinity. So, we can apply L'Hopital's Rule again to find the derivative:

Apply L'Hopital's Rule again

Let's apply L'Hopital's Rule again: $$ \lim_{x\rightarrow\infty}\frac{\beta(\log x)^{\beta-1}}{x}= \lim_{x\rightarrow\infty}\frac{(\beta-1)\beta(\log x)^{\beta-2}}{x} $$ Note that the exponent got reduced to \((\beta - 2)\).

The General Formula and Final Result

At this point, we can observe a pattern. If we kept applying L'Hopital's Rule, we would decrease the exponent of \((\log x)\) until it is negative. When the exponent becomes negative, the function \(\beta(\log x)^k\) will approach 0 as \(x\) approaches infinity since the numerator will be smaller than the denominator. $$ \lim_{x\rightarrow\infty}\frac{(\log x)^{\beta}}{x} = 0 $$ Therefore, the limit of the given function as \(x\) approaches infinity is 0.

Key concepts

L'Hopital's Rule

When tackling limits in real analysis, a common obstacle is the indeterminate form. This is where L'Hopital's Rule comes to the rescue. It serves as a powerful tool to solve limits that involve an indeterminate form such as '0/0' or 'infinity over infinity'. To use L'Hopital's Rule,

you must first verify that you're dealing with an indeterminate ratio. Once confirmed, you take the derivative of the numerator and the derivative of the denominator separately. After this, you re-evaluate the limit of your new function. If the result is still indeterminate, you may apply L'Hopital's Rule repeatedly until you reach a determinate form or identify a pattern that leads to a solution.

Let's remember the key steps to follow when applying L'Hopital's Rule:

• Ensure the limit is in an indeterminate form.
• Differentiate the numerator and denominator independently.
• Re-evaluate the limit, and if necessary, apply L'Hopital's Rule again.

By executing these steps carefully, you can simplify complex limit problems and find solutions that might not be readily apparent.

Indeterminate Form

In real analysis, an indeterminate form poses a significant challenge. It occurs when the limit of a function does not lead directly to a clear result. The 'infinity over infinity' scenario encountered in the exercise is a prime example of such a form. It is not immediately obvious what happens to the ratio when both the numerator and denominator grow without bounds, as they could grow at different rates.

There are several forms of indeterminate expressions, including '0/0', '0 times infinity', and 'infinity minus infinity'. Each of these represents a unique case where a limit needs additional analysis or manipulation to determine its value.

When faced with an indeterminate form, techniques like L'Hopital's Rule, algebraic manipulation, or even transforming the function can provide clarity and a path to the solution. A deep understanding of these forms is essential for solving complex limits and is a central concept in calculus and further mathematical analysis.

Infinity Over Infinity

The 'infinity over infinity' dilemma within limits can be perplexing because it's not immediately clear how the ratio will behave as both the numerator and denominator approach infinity. The rate of growth for both elements is crucial, and different functions grow at different rates.

When dealing with such a problem, the first step is to realize that this scenario is indeed an indeterminate form, one where L'Hopital's Rule can be effectively applied. This rule helps to analyze the behavior of ratios as variables tend towards infinity, often simplifying or transforming the original limit into one that is easier to evaluate.

Understanding how to handle 'infinity over infinity' is more than just applying a rule; it is about interpreting the behavior of functions as they grow without bounds. It is essential to compare the growth rates of the numerator and the denominator and then determine how their ratio behaves as the variable approaches infinity.

Logarithmic Limits

Logarithmic functions are prevalent in limits, especially when they are involved in ratios or as part of exponents, as seen in the given exercise. Considering logarithmic limits involves understanding the behavior of logarithms as the input either approaches zero, infinity, or any other critical value.

In the context of 'infinity over infinity', logarithmic limits can be particularly tricky to evaluate. Logarithmic functions grow very slowly compared to polynomial or exponential functions. As a result, when a logarithmic function is in the numerator, and the denominator is, for example, a linear function like 'x', the denominator grows much faster, and the overall limit tends to zero, as demonstrated in our provided exercise.

Understanding the properties of logarithms, such as their slow growth rate and how they transform multiplication into addition or powers into products, is vital for solving limits. In summary, grasping logarithmic limits eases the way towards evaluating complex limits that involve these functions.