Textbook Question
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
88. lim(x→0) (tan x)/(x + sin(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 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.PE.85
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
85. lim(x→1) (x² + 3x - 4)/(x - 1)
Verified step by step guidance1
First, check if the limit results in an indeterminate form by substituting \(x = 1\) into the expression \(\frac{x^{2} + 3x - 4}{x - 1}\). If both numerator and denominator approach 0, then l'Hôpital's Rule can be applied.
Since direct substitution gives \(\frac{1^{2} + 3(1) - 4}{1 - 1} = \frac{0}{0}\), which is an indeterminate form, proceed with l'Hôpital's Rule.
Apply l'Hôpital's Rule by differentiating the numerator and denominator separately with respect to \(x\). The derivative of the numerator \(x^{2} + 3x - 4\) is \(2x + 3\), and the derivative of the denominator \(x - 1\) is \(1\).
Rewrite the limit using these derivatives: \(\lim_{x \to 1} \frac{2x + 3}{1}\).
Finally, evaluate the new limit by substituting \(x = 1\) into the simplified expression \(\frac{2(1) + 3}{1}\) to find the 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 results in 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
The derivative measures the instantaneous rate of change of a function. To apply l’Hôpital’s Rule, you must differentiate the numerator and denominator separately and then evaluate the new limit.
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Related Practice
Textbook Question
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
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