limits! {cal3}

• limit laws{cal3} (for formal EQs)

Q1:

• exactly 1 of the following limits must exist: (by finding limits along both cruves which disagree w/ e/o (discont)

1. ${\lim }_{(x,y)\to (0,0)}\frac{x^{2}y}{x-y}$
2. ${\lim }_{(x,y)\to (0,0)}\frac{x^2-y^2}{x-y}$

for 1)

• $r_1(t)=t<2,1>,t_0=0$
• $r_2(t)=<t^2,t>,t_0=0$
• Then we plug in $r_1,r_2$ for $(x,y)$ into the limit and solve. We still obtain some DNE, and hence can’t solve

for 2)

• We can just simplify the $f(x,y)$ and we simply obtain 0!

${\lim }_{(x,y)\to (0,0)}f(x,y)$

• When exists, all lim of $f(x,y)$ along r(t) @ (0,0) must exist & agree
• ! exist, 2 curves $r_{1(t)},r_{2(t)}$ s/t lim along 1 $r(t)\ne r_i(t)$

1. Disprove functions follow limits (find @ param curve that approcahes (0,0))

Find 2 @params, if they disagree

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Questions

• You said that 3d calc boils down to taking 1 d methods, what about nth dimensional calc? Would the same tech apply? (to generalize or are the unique n tricks at n dimensions)
• is partial derative shortcut applactiable to nth dimensions?
• What about limits of multiple functions of a matrix? (even without multiple functions)
  • something markov chains? Or some predictive algo?

Vector projection = $\frac{dot}{mag}$ scalar is mag over vectror projection