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__Note:__<= means less than or equal to, >= means greater than or equal to

**1) Prove formally that**

lim xy^2 / (x^2 + y^2) = 0

(x,y)->(0,0)

lim xy^2 / (x^2 + y^2) = 0

(x,y)->(0,0)

*without*breaking the vectors into components.Let X=(x,y). We want to prove that for all epsilon>0, there exists a delta>0 such that if 0<||X-0||=sqrt(x^2 + y^2)<delta, then |xy^2 / (x^2 + y^2)|<epsilon

Starting with |xy^2 / (x^2 + y^2)| <= ? For here HOW can I find an appropriate inequality that can help me find a delta that works? Remember that here we are not allowed to break the vector into individual components.

**2) Evaluate lim e^(x+z) / {z^2 + cos[sqrt(xy)]}.**

(x,y,z)->(1,0,-1)

Solution:

We must have xy>=0

By continuity, the limit equals (direct substitution)

e^(1-1) / {(-1)^2 + cos[sqrt(0)]} = 1/2

(x,y,z)->(1,0,-1)

Solution:

We must have xy>=0

By continuity, the limit equals (direct substitution)

e^(1-1) / {(-1)^2 + cos[sqrt(0)]} = 1/2

I have no idea why they can use continuity/direct subtitution to eavluate the limit...

I believe that as (x,y,z) approachs (1,0,-1) from

**ALL**direction, the limit must exist and equal , but in the case, say, I approach (1,0,-1) through the route (0.99, -0.01, -0.99) ->(0.999, -0.001, -0.999) is clearly

**impossible**because xy=(0.999)(-0.011)<0 which means it is out of the domain. Then how can the limit still exist?

As far as I know, at least some neighbourhood of (1,0,-1) has to be defined for the possibility of a limit to exist...

I am still terribly confused no matter how many times I read the definitions and theorems from my textbook. Any help or explanation will be greatly appreciated!

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