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Chapter 4: Problem 94

Use analytical methods to evaluate the following limits. $$\lim _{x \rightarrow \infty}\left(\log _{2} x-\log _{3} x\right)$$

Short Answer

Expert verified
Based on the step by step solution, the limit as x approaches infinity for the given expression, \(\lim _{x \rightarrow \infty}(\log _{2} x - \log _{3} x)\), is equal to \(\infty\).

Step by step solution

01

Apply change of base formula

First, we will change the base of both logarithms to a common base (natural logarithm), using the change of base formula: $$\lim _{x \rightarrow \infty} \left(\frac{\ln x}{\ln 2} - \frac{\ln x}{\ln 3}\right)$$
02

Factor out common term

Observe that both terms have a common factor of \(\ln x\). We can factor it out of the expression: $$\lim _{x \rightarrow \infty} \ln x \cdot \left(\frac{1}{\ln 2} - \frac{1}{\ln 3}\right)$$
03

Evaluate separately

As there are no variables (\(x\)) in the term within the parenthesis, we can just evaluate it directly: $$\frac{1}{\ln 2} - \frac{1}{\ln 3} = \frac{\ln 3 - \ln 2}{\ln 2 \ln 3} = \frac{\ln(\frac{3}{2})}{\ln 2 \ln 3}$$ Now our expression is: $$\lim _{x \rightarrow \infty} \ln x \cdot \frac{\ln(\frac{3}{2})}{\ln 2 \ln 3}$$
04

Observe and evaluate limit

We see that as \(x\) approaches infinity, the value of \(\ln x\) also increases towards infinity. Meanwhile, the other term, \(\frac{\ln(\frac{3}{2})}{\ln 2 \ln 3}\), is a constant. Therefore, this limit evaluates to: $$\lim _{x \rightarrow \infty}\left(\log _{2} x-\log _{3} x\right) = \infty \cdot \frac{\ln(\frac{3}{2})}{\ln 2 \ln 3} = \infty$$

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