Delta epsilon proof of a continuous limit

• Mar 29th 2013, 04:46 PM
mathimaticool
Delta epsilon proof of a continuous limit
Hi I was just wondering for some help on proofing that this limit exists with the δ-ε proof

Lim (x,y) -> (-1,0). X(x-1)y^2 / (x-1)^2 + y^2

thankyou, I have tried it, but not sure if doing it right
• Mar 29th 2013, 05:42 PM
Prove It
Re: Delta epsilon proof of a continuous limit
To prove $\displaystyle \lim_{(x,y) \to (-1, 0)} \frac{x(x-1)y^2}{(x-1)^2 + y^2} = 0$ you need to show $\displaystyle \sqrt{ (x + 1)^2 + y^2 } < \delta \implies \left| \frac{x(x-1)y^2}{(x-1)^2 + y^2} - 0 \right| < \epsilon$

Can you show us what you've tried please?
• Mar 30th 2013, 06:09 AM
mathimaticool
Re: Delta epsilon proof of a continuous limit
what was done above and ->

y^2 <(or equal to) (x+1)^2 + y^2 since x > -1

so, lx(x-1)l y^2 / (x-1)^2 + y^2 <(or equal too) lx(x+1)l = sqrt((x(x+1))^2) <(or equal too) sqrt((x+1)^2+ y^2)

therefore, δ=ε

Hence, Lim (x,y) -> (-1,0). X(x-1)y^2 / (x-1)^2 + y^2 = 0

is that right?

thankyou