Textbook Question
Evaluate the integrals in Exercises 33–54.
∫(from ln4 to ln9)e^(x/2)dx
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 7.5.32
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
32. lim (x → 0) (3^x - 1) / (2^x - 1)
Verified step by step guidance1
First, recognize that as \(x \to 0\), both the numerator \(3^x - 1\) and the denominator \(2^x - 1\) approach 0, creating an indeterminate form \(\frac{0}{0}\). This is a perfect situation to apply l'Hôpital's Rule.
Recall l'Hôpital's Rule states that if \(\lim_{x \to a} \frac{f(x)}{g(x)}\) results in \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then the limit can be found by evaluating \(\lim_{x \to a} \frac{f'(x)}{g'(x)}\), provided this latter limit exists.
Differentiate the numerator and denominator separately with respect to \(x\). For the numerator, \(f(x) = 3^x - 1\), use the derivative formula for exponential functions: \(\frac{d}{dx} a^x = a^x \ln(a)\). So, \(f'(x) = 3^x \ln(3)\).
Similarly, differentiate the denominator \(g(x) = 2^x - 1\) to get \(g'(x) = 2^x \ln(2)\).
Now, apply l'Hôpital's Rule by taking the limit of the ratio of derivatives as \(x \to 0\): \(\lim_{x \to 0} \frac{3^x \ln(3)}{2^x \ln(2)}\). Since \(3^0 = 1\) and \(2^0 = 1\), substitute these values to simplify the limit expression.
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits and Indeterminate Forms
Limits describe the behavior of a function as the input approaches a particular value. When direct substitution results in an indeterminate form like 0/0 or ∞/∞, special techniques such as l’Hôpital’s rule are needed to evaluate the limit.
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One-Sided Limits
l’Hôpital’s Rule
l’Hôpital’s rule states that if a limit yields an indeterminate form 0/0 or ∞/∞, the limit of the ratio of the functions equals the limit of the ratio of their derivatives, provided this latter limit exists. It simplifies evaluating tricky limits by differentiating numerator and denominator separately.
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Derivatives of Exponential Functions
The derivative of an exponential function a^x with respect to x is a^x times the natural logarithm of a (ln a). This rule is essential when applying l’Hôpital’s rule to limits involving exponential expressions like 3^x and 2^x.
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Related Practice
Textbook Question
In Exercises 7–10, determine from its graph if the function is one-to-one.
f(x) = 3 - x, x < 0
= 3, x ≥ 0
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
Evaluate the integrals in Exercises 41–60.
55. ∫(from -π/4 to π/4)cosh(tanθ)sec²θ dθ
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Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
122. y = (ln x)^(ln x)
Textbook Question
Evaluate the integrals in Exercises 111–114.
112. ∫₁^(eˣ) (1 / t) dt