Limit of x^2/(1-cosx) as x→0 - Solution without LH Rule

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Homework Help Overview

The problem involves finding the limit of the expression x^2/(1-cosx) as x approaches 0, specifically without using L'Hopital's rule.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss various methods to evaluate the limit, including trigonometric identities, power series expansions, and algebraic manipulation of the expression.

Discussion Status

Several potential approaches have been suggested, but no consensus has been reached on the best method. Participants are actively exploring different mathematical techniques to solve the problem without L'Hopital's rule.

Contextual Notes

The original poster indicates a restriction against using L'Hopital's rule, which shapes the direction of the discussion and the methods being considered.

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Homework Statement



Find the limit as x approaches 0 of x^2/(1-cosx).

Homework Equations



None.

The Attempt at a Solution



I know from L'Hopital's rule that the limit is 2, but I'm not supposed to use L'Hopital's rule to calculate it. What else can I do here?
 
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How about the identity cos(x) = 1 - 2sin^2(x/2)?
 
You could also consider the power series expansion of cos(x) around zero.
 
Multiply both numerator and denominator by 1+ cos(x). Then use the fact that
[tex]lim_{x\rightarrow 0}\frac{x}{sin(x)}= 1[/tex].
 

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