Textbook Question
Consider the lim_x→∞ (√ ax + b) / √cx + d where a, b, c, and d are positive real numbers. Show that l’Hôpital’s Rule fails for this limit. Find the limit using another method.
Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 121a
Exponential growth rates
a. For what values of b > 0 does bˣ grow faster than eˣ as x→∞?
Verified step by step guidance1
To determine for which values of b > 0 the function bˣ grows faster than eˣ as x approaches infinity, we need to compare the growth rates of these two functions.
Consider the ratio of the two functions: (bˣ)/(eˣ). We want to find the values of b for which this ratio approaches infinity as x approaches infinity.
Rewrite the ratio using properties of exponents: (bˣ)/(eˣ) = (b/e)ˣ.
For (b/e)ˣ to grow faster than eˣ, the base (b/e) must be greater than 1. This is because if the base of an exponential function is greater than 1, the function grows exponentially as x increases.
Therefore, for bˣ to grow faster than eˣ, b/e > 1, which implies b > e. Thus, b must be greater than the mathematical constant e (approximately 2.718).
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Key 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) = b^x, where b is a positive constant. These functions exhibit rapid growth as x increases, with the base b determining the rate of growth. For example, if b = 2, the function grows faster than linear functions but slower than higher exponential bases.
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Exponential Functions
The Number e
The number e, approximately equal to 2.718, is a fundamental constant in mathematics, particularly in calculus. It serves as the base for natural logarithms and is significant in modeling continuous growth processes. The function e^x is unique because it is the only function that is its own derivative, making it a critical point of comparison for other exponential functions.
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Growth Rate Comparison
To compare the growth rates of two exponential functions, we analyze their limits as x approaches infinity. Specifically, we can determine when b^x grows faster than e^x by examining the ratio of the two functions. If the limit of (b^x)/(e^x) approaches infinity as x increases, then b^x grows faster than e^x, which occurs when b > e.
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Related Practice
Textbook Question
Exponential growth rates
b. Compare the growth rates of eˣ and eᵃˣ as x→∞ , for a > 0.
Textbook Question
Prove that lim_x→∞ (1 + a/x)ˣ = eᵃ , for a ≠ 0 .
Textbook Question
Concavity of parabolas Consider the general parabola described by the function f(x) = ax² + bx + c. For what values of a, b, and c is f concave up? For what values of a, b, and c is f concave down?