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Chapter 4: Problem 114

Show that \(f(x)=a^{x}\) grows faster than \(g(x)=b^{x}\) as \(x \rightarrow \infty\) if \(1

Short Answer

Expert verified
Answer: In the given case, the function \(f(x) = a^x\) grows faster as \(x \rightarrow \infty\).

Step by step solution

01

Set up the ratio of the two functions

First, let's set up the ratio of the two functions as follows, taking \(f(x)\) divided by \(g(x)\): $$ \frac{f(x)}{g(x)} = \frac{a^x}{b^x} $$ Next, we want to simplify the expression by expressing it as a single exponent.
02

Simplify the expression

We can simplify the expression by using the property of exponents, \(\frac{a^x}{b^x} = (\frac{a}{b})^x\). So now we have: $$ \frac{f(x)}{g(x)} = \left(\frac{a}{b}\right)^x $$ Our next step is to find the limit of this expression as x approaches infinity.
03

Find the limit as x approaches infinity

Now, we will find the limit of \(\left(\frac{a}{b}\right)^x\) as \(x \rightarrow \infty\). Since \(1 < b < a\), we know that \(\frac{a}{b} > 1\). Therefore, as x approaches infinity, the expression \(\left(\frac{a}{b}\right)^x\) will also approach infinity: $$ \lim_{x \rightarrow \infty} \left(\frac{a}{b}\right)^x = \infty $$ This limit shows that as x approaches infinity, the ratio of \(f(x)\) to \(g(x)\) approaches infinity, meaning that \(f(x)\) grows faster than \(g(x)\) as \(x \rightarrow \infty\) when \(1 < b < a\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. For example, in the function \( f(x) = a^x \), \( a \) represents the base and \( x \) is the exponent.

One of the critical characteristics of exponential functions is their rapid growth. As the exponent increases, the function's value grows more quickly than polynomial functions. This is because each increase in the exponent results in the base being multiplied by itself another time.

Growth of Exponential Functions

When analyzing the growth of exponential functions, we typically consider the base. If the base is greater than one, the function will grow as the exponent increases. Conversely, if the base is less than one, the function will decrease. This property is instrumental in comparing the growth rates of different exponential functions, as seen in the given exercise.
Limits at Infinity
In calculus, we often want to understand the behavior of functions as the input becomes very large or approaches infinity. The concept of limits at infinity helps us determine the end behavior of functions.

When we take the limit of an exponential function as \( x \) approaches infinity, we're essentially asking what value the function approaches, if any. If a function \( f(x) \) grows without bound as \( x \) increases, the limit is infinity (\( \infty \)), indicating unbounded growth. Conversely, if \( f(x) \) approaches a specific value, that is the limit at infinity.

Understanding Limits at Infinity with Exponential Functions

In the context of the problem, because the function's base is greater than one, the limit as \( x \) approaches infinity is indeed infinity. This is due to the continuous multiplication by a number greater than one, which leads to a value that becomes arbitrarily large.
Properties of Exponents
The properties of exponents are rules that describe how to manipulate expressions involving exponents in algebra. They are foundational for correctly simplifying and comparing exponential functions.

Some essential properties include:
  • The product of powers property: \( a^m \cdot a^n = a^{m+n} \)
  • The quotient of powers property: \( a^m / a^n = a^{m-n} \) or with different bases, \( a^x / b^x = (a/b)^x \)
  • The power of a power property: \( (a^m)^n = a^{m \cdot n} \)
In the given exercise, the quotient of powers property is applied to compare the growth rates of two exponential functions. This property simplifies the expression of the ratio of \( f(x) \) to \( g(x) \), allowing us to find the limit more readily and thus conclude the comparison of growth rates.

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Most popular questions from this chapter

Population models The population of a species is given by the function \(P(t)=\frac{K t^{2}}{t^{2}+b},\) where \(t \geq 0\) is measured in years and \(K\) and \(b\) are positive real numbers. a. With \(K=300\) and \(b=30,\) what is \(\lim P(t),\) the carrying capacity of the population? b. With \(K=300\) and \(b=30,\) when does the maximum growth rate occur? c. For arbitrary positive values of \(K\) and \(b,\) when does the maximum growth rate occur (in terms of \(K\) and \(b\) )?

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