Thread: Help with multivariable calculus problem

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.