Problem 32 - Thomas Calculus

Question

Use l'Hopital's rule to find the limits in Exercises \(7-50\) . $$ \lim _{x \rightarrow \infty} \frac{\log _{2} x}{\log _{3}(x+3)} $$

Step-by-Step Solution

Answer

The limit is \( \frac{\ln 3}{\ln 2} \), or equivalently \( \log_2 3 \).

Step 1: Identify the Indeterminate Form

First, we need to identify the form of the limit. As \( x \to \infty \), both the numerator \( \log_2 x \) and the denominator \( \log_3(x+3) \) approach infinity. Therefore, the limit is of the form \( \frac{\infty}{\infty} \), which is suitable for applying l'Hopital's Rule.

Step 2: Apply l'Hopital's Rule

l'Hopital's Rule can be applied to limits of the form \( \frac{\infty}{\infty} \). Take the derivative of the numerator and the derivative of the denominator. The derivative of \( \log_2 x \) with respect to \( x \) is \( \frac{1}{x \ln 2} \). The derivative of \( \log_3 (x+3) \) with respect to \( x \) is \( \frac{1}{(x+3) \ln 3} \). Therefore, the limit becomes:\[\lim_{x \to \infty} \frac{1/x \ln 2}{1/(x+3) \ln 3}\]

Step 3: Simplify the Expression

Simplify \( \frac{1/x \ln 2}{1/(x+3) \ln 3} \) by multiplying the numerator and the denominator by \((x \ln 2)((x+3) \ln 3)\). This gives:\[\lim_{x \to \infty} \frac{(x+3) \ln 3}{x \ln 2}\]

Step 4: Evaluate the Limit

To find \( \lim_{x \to \infty} \frac{(x+3) \ln 3}{x \ln 2} \), observe that:\[\lim_{x \to \infty} \frac{(x+3)\ln 3}{x \ln 2} = \lim_{x \to \infty} \left(\frac{x\ln 3}{x \ln 2} + \frac{3 \ln 3}{x \ln 2}\right)\]The first part simplifies to \(\frac{\ln 3}{\ln 2}\) and the second part approaches zero as \( x \to \infty \).Therefore:\[\lim_{x \to \infty} \frac{(x+3) \ln 3}{x \ln 2} = \frac{\ln 3}{\ln 2}\]

Step 5: Final Result

The final limit is \( \frac{\ln 3}{\ln 2} \), which can also be expressed as \( \log_2 3 \) because if \( \frac{\log a}{\log b} = \log_b a \). So the limit is an expression for \( \log_2 3 \), approximately 1.585.

Key Concepts

Indeterminate Forms

In calculus, "indeterminate forms" occur when evaluating limits that do not initially provide clear values as we substitute values into them. A common form of such expressions is \( \frac{\infty}{\infty} \). When both the numerator and denominator approach infinity, like in our exercise, we face an indeterminate form. This means the limit cannot be solved directly, and we need to apply special techniques to evaluate it properly.

Indeterminate forms often arise in expressions involving infinity or zero, and they include forms like:

• \( \frac{0}{0} \)
• \( \frac{\infty}{\infty} \)
• \( 0 \cdot \infty \)
• \( \infty - \infty \)
• \( 0^0 \), \( \infty^0 \), and \( 1^{\infty} \)

Recognizing these forms allows us to apply techniques such as l'Hopital's Rule to resolve them. Understanding these forms is crucial since they require special handling that standard algebraic manipulation cannot address.

Limit Evaluation

Limit evaluation is an important concept in calculus that allows us to determine the behavior of functions as inputs approach a specific value. In this problem, we evaluated the limit as \( x \to \infty \). This type of limit is useful to understand the asymptotic behavior of functions when the inputs grow very large.

To evaluate a limit properly, especially when dealing with indeterminate forms, certain techniques and methods are applied. Among these, l'Hopital's Rule is particularly useful when dealing with \( \frac{\infty}{\infty} \) and \( \frac{0}{0} \) forms. Analysis often begins with recognizing the need for such techniques by identifying the behavior of the expressions involved.

When using l'Hopital's Rule, we differentiate the numerator and the denominator separately, and then re-evaluate the limit. This approach simplifies complex scenarios into more manageable ones, making it easier to find the value or behavior of the function as the variable approaches the limit point.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a fundamental concept in calculus, particularly when dealing with growth and decay processes. In our exercise, although we dealt with logarithms of bases other than \( e \), the natural log plays a crucial role due to its property as the derivative of the natural exponential function, \( e^x \).

When calculating limits involving logarithms, converting them to natural logarithms can often simplify derivative calculations. This works because of the change of base formula:

• \( \log_b(x) = \frac{\ln x}{\ln b} \)

In the solution provided, we derive terms like \( \ln 2 \) and \( \ln 3 \) from the change of base formula, simplifying complex logarithmic expressions into tractable forms. This strategy often unveils the straightforward relationships helpful in evaluating limits in calculus.

Calculus Techniques

Various techniques in calculus provide the means to solve limits, derivatives, and integrals of complex functions. In our particular problem, we used l'Hopital's Rule, an incredibly useful tool for dealing with indeterminate forms.

The steps in l'Hopital's Rule involve:

• Identifying that the limit is in an indeterminate form.
• Taking the derivative of the numerator and denominator separately.
• Evaluating the resulting limit of the new function formed by these derivatives.

The power of calculus techniques lies in their structured approach to solving problems that might otherwise seem unsolvable via standard algebraic methods. Techniques like factoring, rationalization, and trigonometric identities also play significant roles in solving different problems by simplifying the expressions into forms amenable to direct calculation or application of rules such as l'Hopital's.

Mastering these techniques allows students to tackle a wide array of problems, providing insights beyond straightforward calculations, expanding the utility of calculus as both a theoretical and applied mathematical tool.