Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→0 csc x sin⁻¹ x
4. Applications of Derivatives
Differentials
2:16 minutesProblem 4.R.89
Textbook Question
82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.
2ˣ and 4ˣ⸍²
Verified step by step guidance1
Step 1: Understand the problem by identifying the two functions given: \( f(x) = 2^x \) and \( g(x) = 4^{x^{1/2}} \). We need to determine which function grows faster as \( x \to \infty \).
Step 2: Simplify the second function \( g(x) = 4^{x^{1/2}} \). Notice that \( 4^{x^{1/2}} = (2^2)^{x^{1/2}} = 2^{2x^{1/2}} \). This means \( g(x) = 2^{2\sqrt{x}} \).
Step 3: Compare the exponents of the base 2 in both functions. For \( f(x) = 2^x \), the exponent is \( x \). For \( g(x) = 2^{2\sqrt{x}} \), the exponent is \( 2\sqrt{x} \).
Step 4: Analyze the growth of the exponents as \( x \to \infty \). The exponent \( x \) in \( f(x) \) grows linearly, while the exponent \( 2\sqrt{x} \) in \( g(x) \) grows as the square root of \( x \), which is slower than linear growth.
Step 5: Conclude that \( f(x) = 2^x \) grows faster than \( g(x) = 4^{x^{1/2}} \) as \( x \to \infty \), because the linear growth of \( x \) in the exponent of \( f(x) \) outpaces the square root growth of \( 2\sqrt{x} \) in the exponent of \( g(x) \).
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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) = a * b^x, where 'a' is a constant, 'b' is the base, and 'x' is the exponent. These functions grow rapidly as 'x' increases, and their growth rate is determined by the base 'b'. In this question, the functions 2^x and 4^x are both exponential, but they have different bases, which affects their growth rates.
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Growth Rate Comparison
To compare the growth rates of two functions, we often analyze their limits or use derivatives. For exponential functions, a common method is to express them in terms of the same base. In this case, 4^x can be rewritten as (2^2)^x = 2^(2x), allowing for a direct comparison with 2^x. This helps in determining which function grows faster as 'x' approaches infinity.
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Limit Analysis
Limit analysis involves evaluating the behavior of functions as they approach a certain point, often infinity. By calculating the limit of the ratio of two functions, we can determine their relative growth rates. If the limit approaches zero, one function grows slower; if it approaches infinity, the other grows faster; and if it approaches a non-zero constant, they grow at comparable rates.
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