Textbook Question
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ ((1/x²) - (2/(x⁵⸍²))) dx
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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.79
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→∞ (1 - (3/x))ˣ
Verified step by step guidance1
Recognize that the limit \( \lim_{x \to \infty} \left(1 - \frac{3}{x}\right)^x \) is of the indeterminate form \(1^\infty\). This suggests the use of the exponential limit property.
Rewrite the expression using the natural exponential function: \( \lim_{x \to \infty} \left(1 - \frac{3}{x}\right)^x = \lim_{x \to \infty} e^{x \ln\left(1 - \frac{3}{x}\right)} \).
Focus on evaluating the exponent \( \lim_{x \to \infty} x \ln\left(1 - \frac{3}{x}\right) \). This is still an indeterminate form \(0 \cdot (-\infty)\).
Use the approximation \( \ln(1 + u) \approx u \) for small \(u\), so \( \ln\left(1 - \frac{3}{x}\right) \approx -\frac{3}{x} \). Substitute this into the limit: \( \lim_{x \to \infty} x \left(-\frac{3}{x}\right) = \lim_{x \to \infty} -3 \).
Conclude that the original limit is \( e^{-3} \) by substituting back into the exponential form: \( \lim_{x \to \infty} e^{x \ln\left(1 - \frac{3}{x}\right)} = e^{-3} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior at points where it may not be explicitly defined, such as at infinity or discontinuities. Evaluating limits is crucial for determining the continuity and differentiability of functions.
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Exponential Functions
Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a positive constant. They exhibit rapid growth or decay and are characterized by their unique property that the rate of change is proportional to the function's value. Understanding their behavior, especially as x approaches infinity, is essential for evaluating limits involving exponential expressions.
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l'Hôpital's Rule
l'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule simplifies the process of finding limits, especially in complex expressions.
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5:50Power Rules
Related Practice
Textbook Question
Change in elevation The elevation h (in feet above the ground) of a stone dropped from a height of 1000 ft is modeled by the equation h(t) = 1000 - 16t², where t is measured in seconds and air resistance is neglected. Approximate the change in elevation over the interval 5 ≤ t ≤ 5.7 (recall that Δh ≈ h' (a) Δt).
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Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_t→0 (1 - cos 6t) / 2t
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Textbook Question
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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Textbook Question
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
c. Give the approximate coordinates of the inflection point(s) of f.
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Textbook Question
24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = 10x² / (x² + 3)
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