Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = (x^2 - 2x + 2)e^(x)
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 7.5.7
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
7. lim (x → 2) (x - 2) / (x² - 4)
Verified step by step guidance1
First, recognize that the limit is of the form \( \frac{0}{0} \) when \( x \to 2 \), since substituting \( x = 2 \) gives \( \frac{2 - 2}{2^2 - 4} = \frac{0}{0} \). This indeterminate form allows us 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 \( \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \), provided the latter limit exists.
Differentiate the numerator and denominator separately: \( f(x) = x - 2 \) so \( f'(x) = 1 \), and \( g(x) = x^2 - 4 \) so \( g'(x) = 2x \).
Rewrite the limit using the derivatives: \( \lim_{x \to 2} \frac{f'(x)}{g'(x)} = \lim_{x \to 2} \frac{1}{2x} \).
Finally, evaluate the new limit by substituting \( x = 2 \) into the expression \( \frac{1}{2x} \), which will give 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 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.
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Derivative of Functions
Derivatives measure the instantaneous rate of change of a function. Applying l’Hôpital’s rule requires finding the derivatives of the numerator and denominator functions to simplify the limit evaluation.
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Related Practice
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.
126. eʸ = y^(ln x)
Textbook Question
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
25. y = ln(ln(x))
1
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Textbook Question
Solve the differential equation in Exercises 9–22.
11. (dy/dx) = e^(x-y)
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Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
59. y = 2^x
Textbook Question
133. Find the absolute maximum value of
f(x) = x^2 * ln(1/x)
and say where it is assumed.