- #1

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## Homework Statement

Examine the behavior of ##f (x,y)= \frac{x^4y^4}{(x^2 + y^4)^3}## as (x,y) approaches (0,0) along various straight lines. From your observations, what might you conjecture ##\lim_{(x,y) \rightarrow (0,0)} f(x,y)## to be? Next, consider what happens when ##(x,y)## approaches ##(0,0)## along the curve ##x=y^2##. Does ##\lim_{(x,y) \rightarrow (0,0)} f(x,y)## exist? Why or why not?

## Homework Equations

## The Attempt at a Solution

Okay, I have no idea what I'm doing but I will try anyway.

As ##x \rightarrow 0## along ##y=0##, ##f(x,y)=\frac{1}{x^2}##

Hence, limit does not exist.

As ##y \rightarrow 0## along ##x=0##, ##f(x,y)= \frac{1}{y^8}##

Hence, limit also does not exist.

When ##x=y^2##,

##f(y^2, y) = \frac{(y^2)^4y^4}{(2y^4)^3}##

##=\frac{1}{8}##

In the first place, I honestly don't understand why the question asked me to consider what happens for the above special case. Since from finding the limit approaching from the ##x## and ##y## axis, I found the limit to not exist, what is the point of doing this additional step? I'm so confused.

And another thing I'm not too sure about. When the textbook asks me to consider the limit when approaching either through the ##x## axis or ##y## axis, does that in any way relate to component functions?

Thanks.