Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
32. lim (x → 0) (3^x - 1) / (2^x - 1)
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.AAE.5
Find the limits in Exercises 1–6.
5. lim(n→∞) (1/(n+1) + 1/(n+2) + ... + 1/(2n))
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Recognize that the expression is a sum of terms of the form \(\frac{1}{k}\) where \(k\) runs from \(n+1\) to \$2n$. This can be written as \(\sum_{k=n+1}^{2n} \frac{1}{k}\).
Recall that the harmonic series \(H_m = \sum_{k=1}^m \frac{1}{k}\) and use it to rewrite the sum as \(H_{2n} - H_n\).
Use the approximation for large \(n\): \(H_n \approx \ln(n) + \gamma\), where \(\gamma\) is the Euler-Mascheroni constant, to express \(H_{2n} - H_n\) as \(\ln(2n) + \gamma - (\ln(n) + \gamma)\).
Simplify the expression by canceling out \(\gamma\) and combining logarithms: \(\ln(2n) - \ln(n) = \ln\left(\frac{2n}{n}\right) = \ln(2)\).
Conclude that the limit as \(n \to \infty\) of the sum is \(\ln(2)\), since the approximation becomes exact in the limit.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit of a Sequence
The limit of a sequence describes the value that the terms of the sequence approach as the index goes to infinity. Understanding how to evaluate limits helps determine the behavior of sequences and series for very large indices.
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8:22
Introduction to Sequences
Harmonic Series and Partial Sums
The harmonic series is the sum of reciprocals of natural numbers. Partial sums of the harmonic series, such as sums from 1 to n, grow without bound but slowly. Recognizing partial sums helps analyze expressions involving sums of terms like 1/k.
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06:45Intro to Series: Partial Sums
Integral Test and Approximation of Sums by Integrals
The integral test relates sums of sequences to definite integrals, allowing approximation of sums by integrals for large n. This technique is useful to estimate sums like 1/(n+1) + ... + 1/(2n) by comparing them to integrals of 1/x.
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07:25Integral Test
Related Practice
Textbook Question
15. Find f'(2) if f(x) = e^(g(x)) and g(x) = ∫(from 2 to x) t/(1+t⁴)dt.
Textbook Question
Theory and Applications
L’Hôpital’s Rule does not help with the limits in Exercises 69–76.
Try it—you just keep on cycling. Find the limits some other way.
71. lim (x → (π/2)⁻) sec x / tan x
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Textbook Question
7. Let A(t) be the area of the region in the first quadrant enclosed by the coordinate axes, the curve y=e^(-x), and the vertical line x=t, t>0. Let V(t) be the volume of the solid generated by revolving the region about the x-axis. Find the following limits.
a. lim(x→∞)A(t)
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Textbook Question
19. Center of mass Find the center of mass of a thin plate of constant density covering the region in the first and fourth quadrants enclosed by the curves y=1/(1+x²) and y=-1/(1+x²) and by the lines x=0 and x=1.
Textbook Question
20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.
b. Find the centroid of the region.
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