Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→∞ (1 - (3/x))ˣ
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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 4.R.83
82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.
x¹⸍² and x¹⸍³
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Step 1: Understand the concept of growth rates. In calculus, when comparing growth rates of functions, we often look at their behavior as x approaches infinity.
Step 2: Consider the functions given: x^(1/2) and x^(1/3). These are both power functions, and their growth rates can be compared by examining their exponents.
Step 3: Recall that for power functions of the form x^a, the function with the larger exponent a will grow faster as x approaches infinity.
Step 4: Compare the exponents of the two functions: 1/2 and 1/3. Since 1/2 is greater than 1/3, the function x^(1/2) grows faster than x^(1/3) as x approaches infinity.
Step 5: Conclude that x^(1/2) grows faster than x^(1/3) based on the comparison of their exponents.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Growth Rates of Functions
The growth rate of a function describes how quickly the function's value increases as the input variable grows. In calculus, we often compare growth rates using limits, derivatives, or asymptotic analysis to determine which function increases faster as x approaches infinity.
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Intro To Related Rates
Limits and Asymptotic Behavior
Limits are fundamental in calculus for analyzing the behavior of functions as they approach a certain point or infinity. Asymptotic behavior refers to the trend of a function as the input becomes very large, allowing us to compare functions by examining their limits at infinity to see which dominates.
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03:07Cases Where Limits Do Not Exist
Power Functions
Power functions are expressions of the form f(x) = x^n, where n is a real number. The growth rate of power functions is determined by the exponent n; specifically, for large values of x, a function with a higher exponent will grow faster than one with a lower exponent, which is crucial for comparing x^(1/2) and x^(1/3).
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04:02The Power Rule
Related Practice
Textbook Question
{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.
c. Compute ƒ'. Then graphƒ and ƒ' for a = 0.5, 1, 2, and 3.
Textbook Question
Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
a. Find the critical points of f and determine where f is increasing and where it is decreasing.
Textbook Question
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) cos² x on [0,π]
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Textbook Question
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
d. Give the approximate coordinates of the zero(s) of f.
Textbook Question
{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.
a. What is the domain of f (in terms of a)?
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