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
Not the one you use?Change textbookSelect a different textbook
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.93
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
93. lim(x→0) (csc(x) - cot(x))
Verified step by step guidance1
First, recognize that as \(x \to 0\), both \(\csc(x)\) and \(\cot(x)\) approach forms that may lead to an indeterminate expression. Specifically, \(\csc(x) = \frac{1}{\sin(x)}\) and \(\cot(x) = \frac{\cos(x)}{\sin(x)}\), so the expression \(\csc(x) - \cot(x)\) can be rewritten as \(\frac{1}{\sin(x)} - \frac{\cos(x)}{\sin(x)}\).
Combine the terms over a common denominator to simplify the expression: \(\csc(x) - \cot(x) = \frac{1 - \cos(x)}{\sin(x)}\).
Evaluate the limit \(\lim_{x \to 0} \frac{1 - \cos(x)}{\sin(x)}\). Direct substitution gives \(\frac{1 - 1}{0} = \frac{0}{0}\), which is an indeterminate form, so l’Hôpital’s Rule applies.
Apply l’Hôpital’s Rule by differentiating the numerator and denominator separately with respect to \(x\): differentiate numerator \(1 - \cos(x)\) to get \(\sin(x)\), and differentiate denominator \(\sin(x)\) to get \(\cos(x)\).
Rewrite the limit as \(\lim_{x \to 0} \frac{\sin(x)}{\cos(x)}\) and then evaluate this new limit by direct substitution.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
l’Hôpital’s Rule
l’Hôpital’s Rule is a method for evaluating limits that result in indeterminate forms like 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.
Recommended video:
Guided course
5:50
Power Rules
Trigonometric Functions and Their Limits
Understanding the behavior of trigonometric functions such as csc(x) and cot(x) near zero is essential. Knowing standard limits like sin(x) ≈ x near zero helps simplify expressions and identify indeterminate forms.
Recommended video:
6:04Introduction to Trigonometric Functions
Derivative of Trigonometric Functions
To apply l’Hôpital’s Rule, one must compute derivatives of csc(x) and cot(x). The derivatives are -csc(x)cot(x) and -csc²(x), respectively, which are crucial for simplifying the limit expression after differentiation.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
85. lim(x→1) (x² + 3x - 4)/(x - 1)
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
1. y = 10e^(-x/5)
Textbook Question
133. What is the age of a sample of charcoal in which 90% of the carbon-14 originally present has decayed?
Textbook Question
In Exercises 79–84, solve for y.
79. 3^y = 2^(y+1)
Textbook Question
113. The function f(x) = e^x + x, being differentiable and one-to-one, has a differentiable inverse f^(-1)(x). Find the value of df^(-1)/dx at the point f (ln 2).