Lim x -> 0 of (1/(2+x) - 1/2)/x

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Homework Help Overview

The discussion revolves around evaluating the limit as x approaches 0 of the expression (1/(2+x) - 1/2)/x. The subject area is calculus, specifically focusing on limits and simplification techniques.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to simplify the expression but seeks alternative methods to solve the limit without using L'Hôpital's rule. Some participants suggest using partial fractions as a potential approach for simplification.

Discussion Status

Participants are exploring different methods for simplifying the limit expression. While one participant offers a specific answer, others are engaged in discussing the simplification process and the use of partial fractions without reaching a consensus on the approach.

Contextual Notes

There is a mention of avoiding L'Hôpital's rule, indicating a preference for algebraic manipulation. The discussion also reflects uncertainty about the correctness of the simplifications presented.

IntegrateMe
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lim x tends to 0 of:

[1/(2+x) - 1/2] / x

I simplified the fraction and got:

[x / 2(2+x)] / x

and with further simplification:

x / x(4+2x)

How do I solve this without using L'Hopital's rule?
 
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The answer is 1, by the way.
 
You can use "partial fractions" to simplify.
[1/(2+x) - 1/2] / x <=> 1/x(2+x) - 1/2x <=> 1/2x-1/(2x+4)-1/2x <=> -1/(2x+4)

When x goes to 0, the solution is goes towards -1/4 = -1/(2*0+4)

The partial fraction of found by solving
1/x(2+x) = a/x + b/(x+2)
 
You can use "partial fractions" to simplify.
[1/(2+x) - 1/2] / x <=> 1/x(2+x) - 1/2x <=> 1/2x-1/(2x+4)-1/2x <=> -1/(2x+4)

When x goes to 0, the solution is goes towards -1/4 = -1/(2*0+4)

The partial fraction of found by solving
1/x(2+x) = a/x + b/(x+2)
 

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