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Chapter 4: Problem 93

Use analytical methods to evaluate the following limits. $$\lim _{x \rightarrow \infty} \frac{\log _{2} x}{\log _{3} x}$$

Short Answer

Expert verified
Question: Find the limit of the given function as x approaches infinity: $$\lim_{x \rightarrow \infty} \frac{\log_{2} x}{\log_{3} x}$$ Answer: The limit of the function as x approaches infinity is $$\frac{\ln 3}{\ln 2}$$.

Step by step solution

01

Rewrite the logarithms with a common base

To evaluate this limit, we first want to rewrite the given logarithmic function with a common base, which we will choose as e. To do this, we can use the change of base formula: $$\log _{a} b = \frac{\ln b}{\ln a}$$ Applying this formula to the given function, we have: $$\frac{\log _{2} x}{\log _{3} x} = \frac{\frac{\ln x}{\ln 2}}{\frac{\ln x}{\ln 3}}$$
02

Simplify the expression

Now, we can simplify the expression by dividing the two fractions: $$\frac{\frac{\ln x}{\ln 2}}{\frac{\ln x}{\ln 3}} = \frac{\ln x}{\ln 2} \cdot \frac{\ln 3}{\ln x}$$ Notice that the ln x terms will cancel out, leaving us with: $$\frac{\ln x}{\ln 2} \cdot \frac{\ln 3}{\ln x} = \frac{\ln 3}{\ln 2}$$
03

Evaluate the limit

Since we've simplified the expression to a constant, the limit becomes trivial, and does not depend on x anymore: $$\lim _{x \rightarrow \infty} \frac{\log _{2} x}{\log _{3} x} = \lim _{x \rightarrow \infty} \frac{\ln 3}{\ln 2}$$ As the expression does not depend on x, the limit is just the constant value: $$\lim _{x \rightarrow \infty} \frac{\log _{2} x}{\log _{3} x} = \frac{\ln 3}{\ln 2}$$

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Most popular questions from this chapter

Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation \(a(t)=v^{\prime}(t)=g,\) where \(g=-9.8 \mathrm{m} / \mathrm{s}^{2}\). a. Find the velocity of the object for all relevant times. b. Find the position of the object for all relevant times. c. Find the time when the object reaches its highest point. What is the height? d. Find the time when the object strikes the ground. A payload is released at an elevation of \(400 \mathrm{m}\) from a hot-air balloon that is rising at a rate of \(10 \mathrm{m} / \mathrm{s}\).

Determine whether the following properties can be satisfied by a function that is continuous on \((-\infty, \infty) .\) If such a function is possible, provide an example or a sketch of the function. If such a function is not possible, explain why. a. A function \(f\) is concave down and positive everywhere. b. A function \(f\) is increasing and concave down everywhere. c. A function \(f\) has exactly two local extrema and three inflection points. d. A function \(f\) has exactly four zeros and two local extrema.

Locate the critical points of the following functions and use the Second Derivative Test to determine whether they correspond to local maxima, local minima, or neither. $$p(t)=2 t^{3}+3 t^{2}-36 t$$

Locate the critical points of the following functions and use the Second Derivative Test to determine whether they correspond to local maxima, local minima, or neither. $$f(x)=x^{3}+2 x^{2}+4 x-1$$

Give an argument to support the claim that if a function is concave up at a point, then the tangent line at that point lies below the curve near that point.
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