Question $4$

Consider the function $(,)=32+4$

$,$ where $$ in $$

is a fixed integer parameter $($we only take integer values to avoid the problem of definability when $<0$

$).$ For which values of $$ in $$

does the limit along any path $=$

is equal to $0?$

In other words, find all $$ in $$

for which we have that

for all $$ in $$

$,$ lim$(,)->(0,0)=(,)=$lim$->0(,)=0.$

$1.$

For $>1$

$2.$

For $>1$

$3.$

For $$ in $\{1,0,1\}$

$4.$

For $<1$

$5.$

For $<1$

$6.$

For no $$ in $$

is the limit at $(0,0)$

along every path $=$

equal to $0.$

$($Hint: As usual, plug in $=$

and try to cancel as high power of $$

as possible. Also, plotting graphs for various values of $$

can be very useful, as well. But, keep in mind that sometimes, due to numerical instability or scaling factor, a plot might be misleading and you can only use it for hints and inspiration, not as a proof. Also, the finer the meshgrid, the more accurate the plot is$,$ although that comes at the cost of more computations, and thus, more time.$)$

Write your answer in the following cell as

ans $=$

For example, if you think the