Textbook Question
Evaluate the integrals in Exercises 39–56.
41. ∫2y dy/(y²-25)
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 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 7.5.39
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
39. lim (x → ∞) (ln 2x - ln(x + 1))
Verified step by step guidance1
First, rewrite the limit expression to understand its form: \(\lim_{x \to \infty} (\ln(2x) - \ln(x + 1))\).
Combine the logarithms using the property \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\), so the limit becomes \(\lim_{x \to \infty} \ln \left( \frac{2x}{x + 1} \right)\).
Focus on the argument of the logarithm: \(\frac{2x}{x + 1}\). To find the limit of the logarithm, first find \(\lim_{x \to \infty} \frac{2x}{x + 1}\).
Since direct substitution leads to an indeterminate form \(\frac{\infty}{\infty}\), apply l’Hôpital’s Rule by differentiating numerator and denominator separately: differentiate \$2x\( and \)x + 1\( with respect to \)x$.
After finding the limit of the fraction using l’Hôpital’s Rule, substitute this limit back into the logarithm to find the overall limit.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas 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
Properties of Logarithms
Understanding logarithmic properties, such as ln(a) - ln(b) = ln(a/b), helps simplify expressions before applying limit techniques. This simplification can make it easier to identify indeterminate forms or apply l’Hôpital’s Rule effectively.
Recommended video:
05:36Change of Base Property
Limits at Infinity
Evaluating limits as x approaches infinity involves analyzing the behavior of functions for very large values of x. Recognizing dominant terms and growth rates is essential to determine whether the limit converges, diverges, or forms an indeterminate expression.
Recommended video:
03:07Cases Where Limits Do Not Exist
Related Practice
Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = ln(e^(θ)/(1+e^θ))
Textbook Question
Evaluate the integrals in Exercises 39–56.
52. ∫(from π/4 to π/2)cot(t)dt
1
views
Textbook Question
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = (x + 3) / (x − 2)
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
37. y=s√(1-s²) + arccos(s)
Textbook Question
130. Use the identity arccot(u)=π/2 - arctan(u) to derive the formula for the derivative of arccot(u) in Table 7.4 from the formula for the derivative of arctan(u).