Question

limit as delta(x)>0+ (1/(x+delta(x))-1/x)/ delta(x)

Answer 1

\[\huge \lim_{\Delta x \rightarrow 0^+}\frac{\frac{1}{x+\Delta x}-\frac{1}{x}}{\Delta x}\]Like this nadine? :o

Answer 2

Perfect!

Answer 3

Hmm it appears we have the Limit Definition of a Derivative.
Gotta do some weird fraction math to get through this one :D

Let's ignore the bottom for a second. We want to get a common denominator between our fractions.

Answer 4

Oh correction! Its as delta(x) approaches from the left

Answer 5

So the denominator would be \[x(x + \Delta(x))\]

Answer 6

Looks good c:

Answer 7

I'm just really confused because I tried to simplify it as much as I could, but the answer is -1/x^2

Answer 8

And I can't figure out how I can get the same answer

Answer 9

Hmm it looks like you're on the right track :D Just gotta do some more simplification.
\[\large \frac{x}{x(x+\Delta x)}-\frac{x+\Delta x}{x(x+\Delta x)}\]
Cancel out the x's on top giving us,
\[\large \frac{\Delta x}{x(x+\Delta x)} \qquad \rightarrow \qquad \frac{\Delta x}{x^2+x \Delta x}\]

Answer 10

The whole idea with a problem like this is,
We want to get it to a point where we can simply PLUG IN delta x, and we won't have a problem. We're not quite to that point yet, but we're close.

Answer 11

Ok bringing back the denominator we originally had, gives us this,
\[\huge \lim_{\Delta x \rightarrow 0^-}\frac{\left(\frac{\Delta x}{x^2+x \Delta x}\right)}{\Delta x}\]Understand where we're at so far? :o

Answer 12

Got it so far! :)))

Answer 13

We can rewrite it like this,\[\huge \lim_{\Delta x \rightarrow 0^-}\left(\frac{\Delta x}{\Delta x(x^2+x \Delta x)}\right)\]Anddd we have another nice cancellation! :D

Answer 14

Woops, before we go too far, I made a silly mistake earlier.

Answer 15

See the second fraction, how it has a negative in front of it? I forgot to DISTRIBUTE the negative to BOTH terms on top of the fraction.

Answer 16

So this Delta x should actually be negative.

Answer 17

Ahhhhh, yep. I get it. :) Nice catch.

Answer 18

And that's how the answer would be negative

Answer 19

I presume ...

Answer 20

OH and then you plug it in for delta(x)

Answer 21

And you get -1/x^2

Answer 22

Yay! Thank you so much!

Answer 23

Yesss \:D/ Good!
After you cancel out the deltas we've successfully gotten it to a point where we can simply plug in for delta x.

Yay team!

Answer 24

Ahhhh, this makes so much sense! Thank you!