Textbook Question
19. Show that e^x grows faster as x→∞ than x^n for any positive integer n, even x^1,000,000. (Hint: What is the nth derivative of x^n?)
1
views
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 7.5.19
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
19. lim (θ → π/6) (sin θ - 1/2) / (θ - π/6)
Verified step by step guidance1
Identify the limit expression: \(\lim_{\theta \to \frac{\pi}{6}} \frac{\sin \theta - \frac{1}{2}}{\theta - \frac{\pi}{6}}\).
Check if the limit is an indeterminate form by substituting \(\theta = \frac{\pi}{6}\): since \(\sin \frac{\pi}{6} = \frac{1}{2}\), the numerator becomes \(\frac{1}{2} - \frac{1}{2} = 0\) and the denominator is \(\frac{\pi}{6} - \frac{\pi}{6} = 0\), so the limit is of the form \(\frac{0}{0}\).
Since the limit is an indeterminate form \(\frac{0}{0}\), apply l’Hôpital’s Rule by differentiating the numerator and denominator separately with respect to \(\theta\).
Compute the derivative of the numerator: \(\frac{d}{d\theta} (\sin \theta - \frac{1}{2}) = \cos \theta\), and the derivative of the denominator: \(\frac{d}{d\theta} (\theta - \frac{\pi}{6}) = 1\).
Rewrite the limit using these derivatives: \(\lim_{\theta \to \frac{\pi}{6}} \frac{\cos \theta}{1}\), then evaluate this limit by substituting \(\theta = \frac{\pi}{6}\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
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 particular 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.
Recommended video:
05:50
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. It simplifies evaluating tricky limits.
Recommended video:
Guided course
5:50Power Rules
Derivative of Trigonometric Functions
Understanding the derivatives of sine and cosine functions is essential when applying l’Hôpital’s rule to trigonometric limits. For example, the derivative of sin(θ) is cos(θ), which is used to find the new limit after differentiation.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
Show that increasing functions and decreasing functions are one-to-one. That is, show that for any x₁ and x₂ in I, x₂ ≠ x₁ implies f(x₂) ≠ f(x₁).
Textbook Question
Evaluate the integrals in Exercises 97–110.
101. ∫ (log₁₀x / x) dx
1
views
Textbook Question
Suppose that the differentiable function y = f(x) has an inverse and that the graph of f passes through the point (2, 4) and has a slope of 1/3 there. Find the value of df⁻¹/dx at x = 4.
Textbook Question
11. Show that if positive functions f(x) and g(x) grow at the same rate as x→∞, then f=O(g) and g=O(f).
1
views
Textbook Question
84. Find lim(x→∞) (√(x² + 1) - √x).
1
views