# Thread: limits with fractions (x in denominator)

1. ## limits with fractions (x in denominator)

Hey guys, I'm new here, looked for a thread on this and didn't see any...

So, my problem is that I have to find the limit as h approaches 0 of 8/x^2.
I can get to lim h approaches 0 of [8/(x^2+2xh+h^2)-8/x^2]/h and then I get stuck.

I've been working on this problem for well over an hour now, and I can't seem to figure out what I'm doing wrong. I've tried a ton of different possibilities from there, but I always end up with an h in the denominator somewhere in the limit... Any suggestions on how I can get the h out of the denominator so I can solve this? (Sorry for the terrible formatting, I don't know how to make it look better).

2. ## find the derivative or the limit of the function?

Are you finding the derivative or the limit of the function? It looks like you are computing f'(x) using the definition, so I'll go with that....

$\lim_{h\to0}{{8\over (x+h)^2}-{8\over x^2}\over h}=\lim_{h\to0}{{8 x^2}-{8(x+h)^2}\over x^2h(x+h)^2}$, multiplying the top and bottom by the common denominator $x^2(x+h)^2$

This then equals

$\lim_{h\to0}{8 x^2-(8x^2+16xh+8h^2)\over x^2h(x+h)^2}=\lim_{h\to0}{-16xh-8h^2\over x^2h(x+h)^2}$, dividing out an h:

$=\lim_{h\to0}{-16x-8h\over x^2(x+h)^2}={-16x\over x^2(x)^2}={-16\over x^3}$

3. It's the derivative, my bad. I have a question on the second step. When you have the two fractions on top, and the h on bottom, you find a common denominator for the top, right? Then in the second step, you have the "h" that was underneath all that in the denominator. I though that since it was on the bottom of the fraction (beneath the num and denom on top) that it would multiply out and be on the top of the fraction when all is said and done... Can you explain that a bit more?

Everything else makes a ton of sense though; I think if I get that one step cleared up that I can get it. Thanks a ton for the help!

4. Originally Posted by insomniac17
It's the derivative, my bad. I have a question on the second step. When you have the two fractions on top, and the h on bottom, you find a common denominator for the top, right? Then in the second step, you have the "h" that was underneath all that in the denominator. I though that since it was on the bottom of the fraction (beneath the num and denom on top) that it would multiply out and be on the top of the fraction when all is said and done... Can you explain that a bit more?

Everything else makes a ton of sense though; I think if I get that one step cleared up that I can get it. Thanks a ton for the help!
$\frac{\frac{8}{(x + h)^2}-\frac{8}{x^2}}{h} = \frac{\frac{8x^2}{x^2(x + h)^2} - \frac{8(x + h)^2}{x^2(x + h)^2}}{h}$

$= \frac{\frac{8x^2 - 8(x + h)^2}{x^2(x + h)^2}}{h}$

$= \frac{\frac{8x^2 - 8(x^2 + 2xh + h^2)}{x^2(x + h)^2}}{h}$

$= \frac{\frac{8x^2 - 8x^2 - 16xh - 8h^2}{x^2(x + h)^2}}{h}$

$= \frac{\frac{-16xh - 8h^2}{x^2(x + h)^2}}{h}$

$= \frac{-16xh - 8h^2}{x^2(x + h)^2}\cdot\frac{1}{h}$

$= \frac{h(-16x - 8h)}{x^2(x + h)^2}\cdot\frac{1}{h}$

$= \frac{-16x - 8h}{x^2(x + h)^2}$.

Now take the limit as $h \to 0$.

5. Oh, okay that makes much more sense now. Thanks a ton for your help guys!