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\centerline
{
\bf Solutions to the Actual ``QUIZ'' for Lecture 6
}

{\bf 1.} 
Find the limit if it exists, or show that the
limit does not exist.

$$
\lim_{(x,y) \rightarrow (0,0)} {{x^4+2y^4} \over {3x^4+5y^4}}
\quad .
$$

{\bf Sol. of 1}: When you plug-in $(x,y)=(0,0)$, you get $0/0$. So we must go on. Let's investigate the limit along a typical lile $y=kx$ with slope $k$.


$$
\lim_{(x,y) \rightarrow (0,0) \quad, y=kx} {{x^4+2y^4} \over {3x^4+5y^4}}
=\lim_{(x,y) \rightarrow (0,0) \quad, y=kx} {{x^4+2(kx)^4} \over {3x^4+5(kx)^4}}
=\lim_{x \rightarrow 0} {{x^4+4k^4x^4} \over {3x^4+5k^4x^4}}
$$
$$
=\lim_{x \rightarrow 0} {{x^4(1+4k^4)} \over {x^4(3+5k^4)}}
=\lim_{x \rightarrow 0} {{(1+4k^4)} \over {(3+5k^4)}}=
\frac{1+4k^4}{3+5k^4} \quad .
$$

The value of the limit along a line $y=kx$ {\bf depends} on $k$, meaning that different $k$ give you different numbers, so there is no consensus,
hence the {\bf limit Does NOT exist!}.



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