Can Taylor Series Help Solve This Integral and Limit?

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Discussion Overview

The discussion revolves around solving an integral and a limit, specifically the integral of \( \frac{1}{x^3+x^2+1} \) and the limit \( \lim_{x \to 0} \frac{x \cos(x) - \sin(x)}{x - \sin(x)} \). Participants explore methods to approach these problems, including the use of Taylor series and partial fractions.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Exploratory

Main Points Raised

  • One participant expresses difficulty in solving the integral \( \int \frac{1}{x^3+x^2+1} \, dx \) and seeks assistance.
  • Another participant suggests that the integral may require the use of partial fractions for simplification.
  • A participant mentions the limit but initially fails to provide it clearly, leading to requests for clarification from others.
  • One participant proposes using Taylor series to expand the trigonometric functions in the limit expression, suggesting that this could simplify the problem.
  • Another participant notes that confirming the limit's result with L'Hôpital's rule would require multiple applications.

Areas of Agreement / Disagreement

There is no consensus on how to approach the integral or the limit, with multiple suggestions and methods proposed by different participants. The discussion remains unresolved regarding the best approach to take.

Contextual Notes

Participants express uncertainty about the applicability of L'Hôpital's rule and whether a theorem exists that relates limits solvable by L'Hôpital's rule to those solvable by other means. There are also unresolved questions about factoring the denominator in the integral.

sutupidmath
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integral, and limit help?

well i have this integral, i have tried to solve it but got nowhere.
it is:

[tex]\int\frac{\1{x^3+x^2+1}[/tex]

and the limit is
 
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You should post this under the applicable homework help section.
 
i am terrible with latex, so i will just write it down. Btw it is not a homework.

integ of 1/(x^3+x^2+1)dx

and the limit i am trying to do it without using the l'hopital rule. By the way i also would like to know if there exists any theorem which states that, if the limit of a functions can be calculated using l'hopital rule, than it will be solvable also without using l'hopital rule??

lim{x-->0)(x cos(x)-sin(x) )/( x- sin(x) )

thnx in advance
 
well, no ideas on how to tackle those two problems so far?
 
Even if it's not "officially" homework, for this type of a problem you have a better chance of getting a response in the applicable HW section.
 
what is the limit? you didn't write it down? and the integral using partial fractions i think
 
ice109 said:
what is the limit? you didn't write it down? and the integral using partial fractions i think

well, i wrote the limit down, read post #3.
About the integral i also think that somewhere along the way i have to use partial fractions, but how to go about factoring the denominator.??
 
Regarding the limit, you could expand the trig terms in Taylor series and collect terms. You should find that you'll be able to divide numerator and denominator by a factor of x^3 to obtain a leading term that is your limit as x approaches zero.

Of course, confirm your answer with L'hopital (requires three applications)
 

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