# Limit problem with fractions

1. Sep 10, 2008

### beneakin

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?

2. Sep 10, 2008

### NoMoreExams

You are on the right track, find the common denominator, what do you get?

3. Sep 10, 2008

### beneakin

wow.. i must have done something wrong in that calculation ...just got what i think is the answer first try.. heres 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

4. Sep 10, 2008

### Gib Z

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!

5. Sep 10, 2008

### NoMoreExams

Actually it's not right, you should have x - (x + h) which is not h