Oops. For Q1 it meant to look like this....
Q1: apply the definition of the limit to prove the following limit exists
lim
(x,y)->(0,0)
x^3 + 4x^2y
x^2+2y^2
Hi guys,
I've got a question set for uni and I'm having trouble with it. There are two questions I'm having trouble with. Here they are.
Q1: apply the definition of the limit to prove the following limit exists
lim x^3 + 4x^2y
(x,y)->(0,0) x^2+2y^2
Q3: z(u,v)=f(x(u,v),y(u,v))
x(u,v)=u^2-v^2 y(u,v)=2uv
show that f_{xx} + f_{yy} = (f_{uu} + f_{vv)/} _{(4(u^2+v^2))}
f_{xx being the partial, double derivative with respect to x, }fyy with respect to y etc.
For Q1, I'm not very good with limits. Apparently we need to do something called a epsilon delta equation and when I looked that up, I got really confused. Even when I try evaluating the limit along different paths, e.g. the x axis, y axis, y=mx etc. I get different values so I'm really confused. You obviously can't evaluate the limit at (0,0) because the function is discontinuous but I'm just not sure what to do.
When I evaluate x->0 I get limit=0
When I evaluate y->0 I get limit=x (and I thought if you evaluated x->0 further you'd still get 0 but I'm not sure if this is right)
is the limit 0? Help me
Also, how would I go about the epsilon delta thingy?
As for Q3. I took the hint and worked with the right hand side. I expected to get a neat little expression for the double derivatives with respect to u and v in terms of the double derivatives with respect to x and y but it just didn't happen too well. Here is a screen capture of what I tried doing and what I ended up with. Maybe I made an arithmetic error but I've tried this question dozens of times.
I really need help . All and any is welcome and appreciated.
Thank you.
Just in case a picture helps with Q3...
... where (key in spoiler) ...
Spoiler:
Add the bottom rows and simplify.
_________________________________________
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
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Ohhhh I see now. My problem was I wasn't using the product rule for the second derivatives because I assumed that you could take all the u and v terms out as a constant. Thank you so much for your help!!!! By the way, you made a slight error (not that it matters) but for your fvv, you have 2(fxy.-2v + fyy.2v). The two in front of the brackets should be 2u.
I'm still stumped for question 1. I'm not very good with limits :S