Textbook Question
Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.
x²⁰ ; 1.0001ˣ
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4. Applications of Derivatives
Differentials
1:51 minutesProblem 121b
Textbook Question
Exponential growth rates
b. Compare the growth rates of eˣ and eᵃˣ as x→∞ , for a > 0.
Verified step by step guidance1
To compare the growth rates of \( e^x \) and \( e^{ax} \) as \( x \to \infty \), we first need to understand the behavior of exponential functions. Both functions are exponential, but they have different exponents.
Consider the function \( e^x \). As \( x \to \infty \), \( e^x \) grows exponentially without bound. The base \( e \) is a constant greater than 1, which means the function increases rapidly.
Now, consider the function \( e^{ax} \). Here, \( a \) is a positive constant greater than 0. The exponent \( ax \) means that the rate of growth is scaled by \( a \). If \( a > 1 \), \( e^{ax} \) grows faster than \( e^x \). If \( 0 < a < 1 \), \( e^{ax} \) grows slower than \( e^x \).
To compare the growth rates more formally, we can take the limit of the ratio of the two functions as \( x \to \infty \): \( \lim_{x \to \infty} \frac{e^{ax}}{e^x} = \lim_{x \to \infty} e^{(a-1)x} \).
Evaluate the limit: If \( a > 1 \), \( e^{(a-1)x} \to \infty \), indicating \( e^{ax} \) grows faster. If \( 0 < a < 1 \), \( e^{(a-1)x} \to 0 \), indicating \( e^x \) grows faster. If \( a = 1 \), the limit is 1, indicating both grow at the same rate.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Functions
Exponential functions are mathematical expressions of the form f(x) = a * e^(bx), where 'e' is the base of the natural logarithm, approximately equal to 2.71828. These functions exhibit rapid growth or decay, depending on the sign of 'b'. Understanding their behavior as x approaches infinity is crucial for analyzing growth rates.
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Growth Rate Comparison
When comparing growth rates of functions, we often look at their limits as x approaches infinity. For the functions e^x and e^(ax) where a > 0, we can determine which function grows faster by evaluating the limit of their ratio. This comparison reveals that e^(ax) grows faster than e^x as x becomes very large.
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Limit Analysis
Limit analysis is a fundamental concept in calculus that helps us understand the behavior of functions as they approach a certain point, often infinity. By applying limit techniques, we can rigorously determine the growth rates of functions like e^x and e^(ax) as x approaches infinity, providing insights into their relative rates of increase.
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