How to Simplify This Fraction Limit Problem?

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    Fractions Limit
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Discussion Overview

The discussion revolves around simplifying a limit problem involving fractions, specifically the limit as h approaches 0 of the expression (1/h) multiplied by the difference of two fractions. Participants explore various approaches to tackle the problem and share their calculations and reasoning.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant presents the limit problem and expresses difficulty in simplifying the expression, considering using the conjugate method but finding it complicated.
  • Another participant suggests finding a common denominator for the fractions involved as a potential next step.
  • A different participant claims to have derived a result of 1/x^2 after simplifying the expression, but later questions the correctness of their approach.
  • Another participant provides feedback on the need for rigor in the working steps and confirms that the previous participant's result is correct.
  • One participant challenges the correctness of the derived result, indicating a mistake in the calculation of the numerator.

Areas of Agreement / Disagreement

There is no consensus on the correctness of the calculations presented, as some participants affirm the results while others point out potential errors. The discussion remains unresolved regarding the accuracy of the final answers.

Contextual Notes

Participants express uncertainty about the steps taken in their calculations, particularly in the context of limits and simplifications. There are indications of missing rigor in the justification of steps leading to conclusions.

beneakin
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I've been working on this one for a while now but just can't figure it out

lim h->0 (1/h) (( 1 / (x + h) ) - ( 1 / x ))

my first thought was to figure out (( 1 / (x + h) ) - ( 1 / x )) first by just putting them togeather and then using the congjigate times by one trick but that just made the problem way more complicated

any hints to get me on the right track?

thanks in advance
 
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beneakin said:
I've been working on this one for a while now but just can't figure it out

lim h->0 (1/h) (( 1 / (x + h) ) - ( 1 / x ))

my first thought was to figure out (( 1 / (x + h) ) - ( 1 / x )) first by just putting them togeather and then using the congjigate times by one trick but that just made the problem way more complicated

any hints to get me on the right track?

thanks in advance

You are on the right track, find the common denominator, what do you get?
 
wow.. i must have done something wrong in that calculation ...just got what i think is the answer first try.. here's what happened...

(x-x+h)/((x+h)x)

h / x(x+h)

then times that by 1/h from the beginning of the problem to get

h / h(x^2+h)

which is

1 / x^2

Is this right??

anyways i tried this route on the next problem i have (very similar problem but with square roots)

lim h->0 (1 / h) ( (1/sqrt(x+h)) - (1/sqrt(x)) )

so i did the common factor and then i multiplied that by
( sqrt(x) + sqrt(x+h) / sqrt(x) + sqrt(x+h) )

to get at the numerator which is sqrt(x) - (1/sqrt(x+h)

which i end up with just h in the numerator

but on the denominator i have ( sqrt(x + h) * sqrt(x) ) ( sqrt(x) + sqrt(x+h) )?

thanks
 
Well, the way you set out your working could be a bit more rigorous; for example, you only get to the last line after you take a limit, otherwise the two last lines aren't actually equal. Things like that aside, you are correct. Well done =]

For your next problem, yes your working is correct so far, don't stop now!
 
Actually it's not right, you should have x - (x + h) which is not h
 

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