Question

find the limit if it exists:
lim Δx->0 ((x+Δx)^2-x^2)/(Δx)

Answer 1

i tried factoring out the swuared part but i messed up somewhere, im taking another shot at it

Answer 2

Start by expanding \[(x + Deltax)^{2}\]

Answer 3

(x+Δx)(x+Δx)=x^2+Δx^2+Δx^2+2Δx^2-x^2/Δx. that is what i got. i think it is wrong though.

Answer 4

\[(x + \Delta x)^{2}\] = \[x ^{2} + (\Delta x)^{2} + 2x(\Delta x)\]

Answer 5

the delta x is just a single variable. It looks confusing because it requires more than one symbol.

Answer 6

im going to try again

Answer 7

so the first step was to get: x^2+x(Δx)+x(Δx)+(Δx)^2?

Answer 8

Once you subtract out an x^2 from what I showed you, you'll have 2 terms left that you can factor the delta x out of because of the delta x in the denominator.

Answer 9

by doing that i get 2x(Δx)+Δx^2 / Δx

Answer 10

Yes, that was the first part but not the key. The key is to get rid of the delta x in the denominator.

Answer 11

yes i know. im just taking it step by step. this is my first time seeing this delta symbol being used in this manner

Answer 12

np, you're doing well. So, the numerator can factor out a delta x.

Answer 13

And because you'll have a factor of delta x in both the numerator and denominator, those factors will cancel.

Answer 14

hey, im going to type something out. tell me if this is true

Answer 15

i tried factoring out the numerator and got this. (Δx)(2x+Δx)/(Δx). so i am left with just 2x+Δx.

Answer 16

Very, very good. That's the key. Now there's no denominator to worry about as delta x goes to 0.

Answer 17

yay~!

Answer 18

yay is key! Actually, groovy!

Answer 19

you cleared things up big time by having me look at (Δx) as one thing.

Answer 20

Math is way cool when the lightbulb goes on.

Answer 21

yep. so now 0 is plugged in and i get 2(0)+Δ(0)=0

Answer 22

almost

Answer 23

you are left with 2x + delta x. It is only the delta x that is going to 0 so you are left with 2x once delta x goes to 0

Answer 24

oh wait the lmiit is Δx. so only 0 can replace Δx, correct?

Answer 25

i get ahead of myself sometimes :\

Answer 26

Yes, only delta x is approaching 0. x stays the same.

Answer 27

ahh i see. so the answer would be lim Δx-> = 2x

Answer 28

That's it! and with that, you have the answer!

Answer 29

aweeeeeesome. thank you so much!

Answer 30

thx for the medal and more importantly, good luck to you always in all your studies. Learning can be very fun.

Answer 31

yea, sure is!