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 7.5.71
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
Verified step by step guidance1
Identify the limit expression: \(\lim_{x \to (\pi/2)^-} \frac{\sec x}{\tan x}\).
Rewrite the trigonometric functions in terms of sine and cosine to simplify the expression: \(\sec x = \frac{1}{\cos x}\) and \(\tan x = \frac{\sin x}{\cos x}\). Substitute these into the limit to get \(\lim_{x \to (\pi/2)^-} \frac{\frac{1}{\cos x}}{\frac{\sin x}{\cos x}}\).
Simplify the complex fraction by multiplying numerator and denominator appropriately: \(\frac{\frac{1}{\cos x}}{\frac{\sin x}{\cos x}} = \frac{1}{\cos x} \times \frac{\cos x}{\sin x} = \frac{1}{\sin x}\).
Now the limit reduces to \(\lim_{x \to (\pi/2)^-} \frac{1}{\sin x}\). Consider the behavior of \(\sin x\) as \(x\) approaches \(\pi/2\) from the left side.
Since \(\sin x\) approaches 1 as \(x\) approaches \(\pi/2\), analyze the limit \(\frac{1}{\sin x}\) accordingly to determine the limit value.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Behavior of Trigonometric Functions Near Specific Points
Understanding how functions like sec x and tan x behave as x approaches π/2 from the left is crucial. Near π/2, tan x tends to infinity, while sec x also grows large but their rates differ, affecting the limit's value.
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Guided course
6:04
Introduction to Trigonometric Functions
Limits Involving Indeterminate Forms and Alternative Techniques
When L’Hôpital’s Rule leads to cyclic or unresolvable forms, alternative methods such as algebraic manipulation, trigonometric identities, or analyzing one-sided limits must be used to evaluate the limit.
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03:33Integrals Involving Natural Logs: Substitution Example 7
One-Sided Limits and Directional Approach
Evaluating limits as x approaches π/2 from the left (denoted π/2⁻) requires considering the function's behavior just before π/2, which can differ from approaching from the right, impacting the limit's existence and value.
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05:50One-Sided Limits
Related Practice
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