Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
3. a. arcsin(-1/2)
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.PE.97
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
97. lim(x→0) (10^x - 1)/x
Verified step by step guidance1
Identify the limit expression: \(\lim_{x \to 0} \frac{10^x - 1}{x}\).
Check the form of the limit by substituting \(x = 0\): numerator becomes \(10^0 - 1 = 1 - 1 = 0\) and denominator is \(0\), so the limit is of the form \(\frac{0}{0}\), which is indeterminate and allows the use of l'Hôpital's Rule.
Apply l'Hôpital's Rule by differentiating the numerator and denominator separately with respect to \(x\): differentiate \$10^x\( using the chain rule, recalling that \(\frac{d}{dx} a^x = a^x \ln(a)\) for a positive constant \)a$.
Write the new limit after differentiation: \(\lim_{x \to 0} \frac{\frac{d}{dx}(10^x - 1)}{\frac{d}{dx} x} = \lim_{x \to 0} \frac{10^x \ln(10)}{1}\).
Evaluate the new limit by substituting \(x = 0\): \(10^0 \ln(10) = 1 \cdot \ln(10)\), which gives the value of the original limit.
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Key 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 certain value. When direct substitution results in an indeterminate form like 0/0, 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 is a method for evaluating limits that yield indeterminate forms 0/0 or ∞/∞. It states that the limit of a ratio of functions can be found by taking the limit of the ratio of their derivatives, provided certain conditions are met.
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Derivative of Exponential Functions
The derivative of an exponential function a^x, where a > 0, is given by 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.
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Related Practice
Textbook Question
84.a. Find the center of mass of a thin plate of constant density covering the region between the curve y=1/√x and the x-axis from x=1 to x=16.
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Textbook Question
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
88. lim(x→0) (tan x)/(x + sin(x))
Textbook Question
110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
c. f(x) = 10x^3 + 2x^2, g(x) = e^x
Textbook Question
In Exercises 79–84, solve for y.
79. 3^y = 2^(y+1)
Textbook Question
23. What roles do the functions e^x and ln(x) play in growth comparisons?