multivariable limit: lim cos[(x^3 - y^3)/(x^2 + y^2)]

[cheffy]

Find the limit of f(x,y) as (x,y) -> (0,0)

\(\displaystyle \
f(x,y) = \cos \left( {\frac{{x^3 - y^3 }}{{x^2 + y^2 }}} \right)
\\)

My intuition says that this DNE, but I don't know what path to plug in to prove it. (or my intuition is wrong) Thanks!

 

[pka]

I disagree with your intuition:

Use parametric forms: $\displaystyle x = r\cos (t),\;y = r\sin (t),\;r = \sqrt {x^2 + y^3 }$

Then $\displaystyle \frac{{x^3 - y^3 }}{{x^2 + y^2 }} = \frac{{r^3 \left[ {\cos ^3 (t) - \sin ^3 (t)} \right]}}{{r^2 }} = r\left[ {\cos ^3 (t) - \sin ^3 (t)} \right]$.
Thus $\displaystyle (x,y) \to (0,0)\; \Rightarrow \;r \to 0$ so $\displaystyle \frac{{x^3 - y^3 }}{{x^2 + y^2 }} = r\left[ {\cos ^3 (t) - \sin ^3 (t)} \right] \to 0$.

Now we get $\displaystyle \cos (0) = 1$.

Is it possible that the problem is different from the one posted?