Explanation Given: The provided function is lim x → ∞ ∫ 1 x ln ( e 4 t − 1 ) d t x . Formula used: L’Hospital’s Rule: If the limit of f ( x ) / g ( x ) when x approaches c from right produces indeterminate form 0 / 0 , then: lim x → c f ( x ) g ( x ) = lim x → c f ′ ( x ) g ′ ( x ) Power rule: d d x x n = n x n − 1 Integration formula: ∫ a b x n d x = | x n + 1 n + 1 | a b Calculation: Consider the function lim x → ∞ ∫ 1 x ln ( e 4 t − 1 ) d t x . Recall the integration property and simplify: ∫ a b x n d x = | x n + 1 n + 1 | a b Apply the integration property, lim x → ∞ ∫ 1 x ln ( e 4 t − 1 ) d t x = lim x → ∞ ∫ 1 x ( 4 t − 1 ) d t x = lim x → ∞ | 4 ⋅ t 1 + 1 1 + 1 − t | 1 x x = lim x → ∞ | 4 ⋅ t 2 2 − t | 1 x x = lim x → ∞ | 2 t 2 − t | 1 x x Further put the limits of integration, lim x → ∞ ∫ 1 x ln ( e 4 t − 1 ) d t x = lim x → ∞ | 2 t 2 − t | 1 x x = lim x → ∞

10th Edition

ISBN: 9781285057095

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

Videos

Question

Chapter 8.7, Problem 41E

To determine

To calculate: The value of limit limx→∞∫1xln(e4t−1)dtx by the use of L’Hospital’s Rule if necessary.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.7, Problem 40E

Chapter 8.7, Problem 42E

Students have asked these similar questions

This question is a previous exam question. I am using it for practice but am stuck

in Q. A firm price of 501: If the Total cast is given by perfect competition sells its products at the TTC = 3Q² +2Q+5. level of output will will be the level of profit at What What Devive the Consumer Curve approach. demand the function maximize this firm's, that using putput level. the indifference prpfit. Q₂. The Total Cost equation in the production of bacon has hypothetical factor a 2 A C= "TC 1000+ 159" +03 ; Where ç. Kash, Bacao - metric bone Compute and 11" tonnes the and average cost at output level of 10. Stretch theme marginal cost of the the shope Carve an the production average, Cost arve 12 tonnes and explain, the relationship between Marginal Cost product es tamen op d Galaxy A71 01 Curve in

if w(x, y, z) = sin' ( xyz) (y zî + x z j + xy k) Find grad (div) at (0.5, 1, 0.5) (xyz)2

Chapter 8 Solutions

Calculus

Ch. 8.1 - Choosing an Antiderivative In Exercises 3 and 4,...Ch. 8.1 - Choosing an Antiderivative In Exercises 3 and 4,...Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Choosing a Formula In Exercises 514, select the...Ch. 8.1 - Prob. 6E Ch. 8.1 - Choosing a Formula In Exercises 514, select the...Ch. 8.1 - Prob. 8E Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The value of limit lim x → ∞ ∫ 1 x ln ( e 4 t − 1 ) d t x by the use of L’Hospital’s Rule if necessary.

Solution Summary: The author explains how to calculate the value of limit undersetxto inftymathrmlimdisplaystyle

Concept explainers

Videos

To calculate: The value of limit limx→∞∫1xln(e4t−1)dtx by the use of L’Hospital’s Rule if necessary.

Want to see the full answer?

Chapter 8 Solutions